# questions

title: solution-of-2-x-2-x-2-2-is
slug: solution-of-2-x-2-x-2-2-is-96936
question: <div class="question">Solution of $2^x+2^{|x|} \geq 2 \sqrt{2}$ is</div>
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(-\infty, \log _2(\sqrt{2}+1)\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(0, \infty)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{2}, \log _2(\sqrt{2}-1)\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left(-\infty, \log _2(\sqrt{2}-1)\right] \cup\left[\frac{1}{2}, \infty\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(-\infty, \log _2(\sqrt{2}-1)\right] \cup\left[\frac{1}{2}, \infty\right)$</span> </div>

answer:
<div class="solution">$2^x+2^{|x|} \geq 2 \sqrt{2}$ ...(i)<br/>Case I: $x \geq 0$, then Eq. (i) becomes<br/>$2^x+2^x \geq 2 \sqrt{2}$<br/>$\Rightarrow 2^x \geq \sqrt{2} \Rightarrow x \geq \frac{1}{2}$<br/>Case II: $x &lt; 0$, then Eq. (i) becomes<br/>$2^x+2^{-x} \geq 2 \sqrt{2}$<br/>$\Rightarrow t+\frac{1}{t} \geq 2 \sqrt{2}$, where $2^x=t$<br/>$\Rightarrow t^2-2 \sqrt{2} t+1 \geq 0$<br/>$\Rightarrow x \leq \log _2(\sqrt{2}-1)$<br/>Also, $0 &lt; \sqrt{2}-1 &lt; 1, \log _2(\sqrt{2}-1) &lt; 0$.<br/>$\therefore$ The solution is<br/>$\left.\left(-\infty, \log _2(\sqrt{2}-1)\right)\right] \cup\left[\frac{1}{2}, \infty\right)$</div>


title: solution-of-differential-equation-x-2-1-y-x-1-d-x-d-y-2-y-x-2-d-x-d-y-2-3-y-x-3-d-x-d-y-3-is
slug: solution-of-differential-equation-x-2-1-y-x-1-d-x-d-y-2-y-x-2-d-x-d-y-2-3-y-x-3-d-x-d-y-3-is-74999
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">Solution of differential equation
<br/> <br/>$$
<br/> <br/>x^{2}=1+\left(\frac{x}{y}\right)^{-1} \frac{d y}{d x}+\frac{\left(\frac{x}{y}\right)^{-2}\left(\frac{d y}{d x}\right)^{2}}{2 !}+\frac{\left(\frac{x}{y}\right)^{-3}\left(\frac{d y}{d x}\right)^{3}}{3 !}+\ldots \text { is }
<br/> <br/>$$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$y^{2}=x^{2}\left(\ln x^{2}-1\right)+C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y=x^{2}(\ln x-1)+C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y^{2}=x(\ln x-1)+\mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=x^{2} e^{x^{2}}+\mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y^{2}=x^{2}\left(\ln x^{2}-1\right)+C$</span> </div>

answer:
<div class="solution">$x^{2}=e^{\left(\frac{x}{y}\right)^{-1}\left(\frac{d y}{d x}\right)} \Rightarrow x^{2}=e^{\left(\frac{y}{x}\right)\left(\frac{d y}{d x}\right)}$
<br/> <br/>$\Rightarrow \ln x^{2}=\frac{y}{x} \frac{d y}{d x}$ or $\int x \ln x^{2} d x=\int y d y$
<br/> <br/>Put $x^{2}=t \Rightarrow 2 x d x=d t \therefore \frac{1}{2} \int \ln t d t=\frac{y^{2}}{2}$
<br/> <br/>$C+t \ln t-t=y^{2}$ or $y^{2}=x^{2}\left(\ln x^{2}-1\right)+C$</div>


title: solution-of-the-differential-equation-d-y-d-x-sin-x-y-cos-x-y-is-equal-to
slug: solution-of-the-differential-equation-d-y-d-x-sin-x-y-cos-x-y-is-equal-to-72592
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question">Solution of the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>sec</mi><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>=</mo><mi>y</mi><mo>+</mo><mi>C</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>tan</mi><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math></p><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>v</mi><mo>⇒</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>-</mo><mn>1</mn><mo>=</mo><mi>sin</mi><mi>v</mi><mo>+</mo><mi>cos</mi><mi>v</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>sin</mi><mi>v</mi><mo>+</mo><mi>cos</mi><mi>v</mi><mo>+</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>sin</mi><mi>v</mi><mo>+</mo><mi>cos</mi><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∫</mo><mfenced><mfrac><mn>1</mn><mrow><mi>sin</mi><mi>v</mi><mo>+</mo><mi>cos</mi><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi>d</mi><mi>v</mi><mo>=</mo><mo>∫</mo><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∫</mo><mfrac><mn>1</mn><mrow><mfenced close="}" open="{"><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi>tan</mi><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow></mfrac></mstyle></mfenced><mo>+</mo><mfenced close="}" open="{"><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>tan</mi><mn>2</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow></mfrac></mstyle></mfenced><mo>+</mo><mn>1</mn></mrow></mfrac><mi>d</mi><mi>v</mi><mo>=</mo><mo>∫</mo><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∫</mo><mfenced close="]" open="["><mfrac><mfenced close="}" open="{"><mrow><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow></mfenced><mrow><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>tan</mi></mstyle><mstyle displaystyle="true"><mfenced><mfrac><mstyle displaystyle="true"><mi>v</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></mfenced></mstyle><mo>+</mo><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><mo>-</mo></mstyle><mstyle displaystyle="true"><msup><mstyle displaystyle="true"><mi>tan</mi></mstyle><mn>2</mn></msup></mstyle><mstyle displaystyle="true"><mfenced><mfrac><mstyle displaystyle="true"><mi>v</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></mfenced></mstyle><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>tan</mi><mn>2</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow></mfrac></mfenced><mi>d</mi><mi>v</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfenced close="]" open="["><mfrac><mrow><msup><mi>sec</mi><mn>2</mn></msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mn>2</mn></mfrac></mstyle></mfenced></mrow><mstyle displaystyle="true"><mi>tan</mi><mfenced><mfrac><mstyle displaystyle="true"><mi>v</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></mfenced><mo>+</mo><mn>1</mn></mstyle></mfrac></mfenced><mi>d</mi><mi>v</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>log</mi><mfenced><mrow><mi>tan</mi><mfenced><mfrac><mstyle displaystyle="true"><mi>v</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>log</mi><mfenced><mrow><mi>tan</mi><mfenced><mfrac><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>C</mi></math></p></div>


title: solution-of-the-differential-equation-fracd-yd-xfracyxsin-x-is
slug: solution-of-the-differential-equation-fracd-yd-xfracyxsin-x-is-60055
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
question:
<div class="question">Solution of the differential equation \(\frac{d y}{d x}+\frac{y}{x}=\sin x\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(x(y+\cos x)=\cos x+C\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(x(y-\cos x)=\sin x+C\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(x(y+\cos x)=\sin x+C\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x(y+\cos x)=\sin x+C\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \frac{d y}{d x}+\frac{1}{x} \cdot y=\sin x\left[\text { Type } \frac{d y}{d x}+P y=Q\right] \\<br/>&amp; e^{\int P d x}=e^{\int \frac{1}{x} d x}=e^{\log x}=x<br/>\end{aligned}\)<br/>\(\therefore\) Sol. is \(\mathrm{yx}=\int \mathrm{x} \sin \mathrm{xdx}+\mathrm{C}\)<br/>\(\begin{aligned}<br/>&amp; =x(-\cos x)-\int 1 \cdot(-\cos x) d x+C \\<br/>&amp; =-x \cos x+\sin x+C \Rightarrow x(y+\cos x)=\sin x+C<br/>\end{aligned}\)</div>


title: solution-of-the-equation-d-y-d-x-e-x-y-e-x-e-y-is
slug: solution-of-the-equation-d-y-d-x-e-x-y-e-x-e-y-is-84458
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">Solution of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></msup><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>e</mi><mi>y</mi></msup></mrow></mfenced></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>y</mi></msup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn><mo>+</mo><mi>c</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></msup><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mi>c</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>x</mi></msup><mo>+</mo><msup><mi>e</mi><mi>y</mi></msup><mo>=</mo><mi>c</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>y</mi></msup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn><mo>+</mo><mi>c</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></msup><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>e</mi><mi>y</mi></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>e</mi><mi>y</mi></msup><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>+</mo><msup><mi>e</mi><mi>y</mi></msup><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></math></p><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mi>y</mi></msup><mo>=</mo><mi>v</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>e</mi><mi>y</mi></msup><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><mi>d</mi><mi>v</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mi>v</mi><msup><mi>e</mi><mi>x</mi></msup><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup></math></p><p>This is linear equation </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>.</mo><mi mathvariant="normal">F</mi><mo>=</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>∫</mo><msup><mi>e</mi><mi>x</mi></msup><mi>d</mi><mi>x</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Required solution is </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>·</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup><mo>=</mo><mo>∫</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup><mo>·</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>·</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup><mo>=</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>e</mi><mi>y</mi></msup><mo>·</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup><mo>=</mo><msup><mi>e</mi><msup><mi>e</mi><mi>x</mi></msup></msup><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mi>C</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>e</mi><mi>y</mi></msup><mo>=</mo><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn><mo>+</mo><mi>c</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></math></p></div>


title: solve-5-x-2-x-5-0
slug: solve-5-x-2-x-5-0-40974
question: <div class="question">Solve $\sqrt{5} x^2+x+\sqrt{5}=0$</div>
tags: ['Mathematics', 'Quadratic Equation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pm \frac{\sqrt{19}}{5} \mathrm{i}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pm \frac{\sqrt{19} i}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{-1 \pm \sqrt{19} i}{2 \sqrt{5}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{-1 \pm \sqrt{19} \mathrm{i}}{\sqrt{5}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-1 \pm \sqrt{19} i}{2 \sqrt{5}}$</span> </div>

answer:
<div class="solution">Here, $\mathrm{b}^2-4 \mathrm{ac}=1^2-4 \times \sqrt{5} \times \sqrt{5}$ $=1-20=-19$<br/>Therefore, the solutions are<br/>$\frac{-1 \pm \sqrt{-19}}{2 \sqrt{5}}=\frac{-1 \pm \sqrt{19 i}}{2 \sqrt{5}}$</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
some-amount-of-a-radioactive-substance-halflife-10-days-is-spread-inside-a-room-and-consequently-the-level-of-radiation-becomes-50-times-the-permissib-1

slug:
some-amount-of-a-radioactive-substance-halflife-10-days-is-spread-inside-a-room-and-consequently-the-level-of-radiation-becomes-50-times-the-permissib-1-37884

question:
<div class="question">Some amount of a radioactive substance (half-life <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10</mn></math> days ) is spread inside a room and consequently, the level of radiation becomes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> times the permissible level for normal occupancy of the room. After how many days will the room be safe for occupation?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math> days</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>34</mn><mo>.</mo><mn>8</mn></math> days</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>56</mn><mo>.</mo><mn>4</mn></math> days</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>62</mn><mo>.</mo><mn>9</mn></math> days</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>56</mn><mo>.</mo><mn>4</mn></math> days</span> </div>

answer:
<div class="solution"><p>since, the initial activity is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> times the activity for safe occupancy, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>0</mn></msub><mo>=</mo><mn>50</mn><mi>R</mi><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>λ</mi><mi>N</mi></math>.<br/>since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>∝</mo><mi>N</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><msub><mi>R</mi><mn>0</mn></msub></mfrac><mo>=</mo><mfrac><mi>N</mi><msub><mi>N</mi><mn>0</mn></msub></mfrac><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mo>/</mo><mi>T</mi></mrow></msup></math></p><p>Or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mo>/</mo><mn>10</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mn>50</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mn>2</mn></mfenced><mrow><mi>t</mi><mo>/</mo><mn>10</mn></mrow></msup><mo>-</mo><mn>50</mn><mo>⇒</mo><mfrac><mi>t</mi><mn>10</mn></mfrac><msub><mi>log</mi><mn>10</mn></msub><mn>2</mn><mo>=</mo><msub><mi>log</mi><mn>10</mn></msub><mn>50</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mi>t</mi><mo>=</mo><mfrac><mrow><mn>10</mn><msub><mi>log</mi><mn>10</mn></msub><mn>50</mn></mrow><msubsup><mi>log</mi><mn>10</mn><mn>2</mn></msubsup></mfrac><mo>=</mo><mfrac><mrow><mn>10</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>699</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>301</mn></mrow></mfrac><mo>=</mo><mn>56</mn><mo>.</mo><mn>4</mn></math> days</p></div>


tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
title:
some-salts-containing-two-different-metallic-elements-give-test-for-only-one-of-them-in-solution-such-salts-are

slug:
some-salts-containing-two-different-metallic-elements-give-test-for-only-one-of-them-in-solution-such-salts-are-79252

question:
<div class="question">Some salts containing two different metallic elements give test for only one of them in solution, such salts are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">double salts</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">normal salts</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">complex salts</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">complex salts</span> </div>

answer:
<div class="solution">Complex compounds contains two different metallic elements but give test only for one of them. Because complex ions such as [Fe $\left.(\mathrm{CN})_{6}\right]^{4-}$ of $\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right],$ do not dissociate
<br/> <br/>into Fe $^{2+}$ and $\mathrm{CN}$ -ions.</div>


title: sometimes-the-colour-observed-in-lassaignes-test-for-nitrogen-is-green-it-is-because
slug: sometimes-the-colour-observed-in-lassaignes-test-for-nitrogen-is-green-it-is-because-72492
tags: ['Chemistry', 'Practical Chemistry', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">Sometimes, the colour observed in Lassaigne's test for nitrogen is green. It is because</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">of green colour of ferrous sulphate</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">ferric ferrocyanide is also green</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">of green colour of copper sulphate</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">of excess of Fe $^{3+}$ ions whose yellow colour makes the blue colour of ferric ferrocyanide to appear green.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">of excess of Fe $^{3+}$ ions whose yellow colour makes the blue colour of ferric ferrocyanide to appear green.</span> </div>

answer:
<div class="solution">Although blue coloured ferric ferrocyanide is formed but due to the presence of yellow coloured $\mathrm{Fe}^{3+}$ salts, the blue colour gives the shade of green.</div>


title: spin-only-magnetic-moment-of-mn-4-ion-is-nearly
slug: spin-only-magnetic-moment-of-mn-4-ion-is-nearly-67531
tags: ['Chemistry', 'd and f Block Elements', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">Spin only magnetic moment of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Mn</mi><mrow><mn>4</mn><mo>+</mo></mrow></msup></math> ion is nearly</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo> </mo><mi>BM</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi>BM</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>BM</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><mi>BM</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>BM</mi></math></span> </div>

answer:
<div class="solution"><p>Electronic configuration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Mn</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">Z</mi><mo>=</mo><mn>25</mn></mrow></mfenced><mo>=</mo><mfenced close="]" open="["><mi>Ar</mi></mfenced><mn>3</mn><msup><mi mathvariant="normal">d</mi><mn>5</mn></msup><mn>4</mn><msup><mi mathvariant="normal">s</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Electronic configuration of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Mn</mi><mrow><mn>4</mn><mo>+</mo></mrow></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="]" open="["><mi>Ar</mi></mfenced><mn>3</mn><msup><mi mathvariant="normal">d</mi><mn>3</mn></msup></math></p><p>Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Mn</mi><mrow><mn>4</mn><mo>+</mo></mrow></msup></math> has <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>-unpaired electron and its spin only magnetic moment <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">μ</mi></mfenced></math> is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">μ</mi><mo>=</mo><msqrt><mi mathvariant="normal">n</mi><mfenced><mrow><mi mathvariant="normal">n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></msqrt></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>BM</mi></math></p><p>where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">n</mi><mo>=</mo></math> number of unpaired electrons.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">μ</mi><mo>=</mo><msqrt><mn>3</mn><mo>(</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>)</mo></msqrt><mo>.</mo><mo> </mo><mi>BM</mi><mo>⇒</mo><mi mathvariant="normal">μ</mi><mo>=</mo><msqrt><mn>15</mn></msqrt><mo>=</mo><mn>3</mn><mo>.</mo><mn>87</mn><mi>BM</mi><mo>≈</mo><mn>4</mn><mi>BM</mi></math></p></div>


tags: ['Mathematics', 'Pair of Lines', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
title:
st-a-t-e-m-e-n-t-1-if-the-general-equation-x-2-y-2-2-x-y-2-gx-2-f-y-4-0-represents-a-pair-of-real-lines-then-g-2-st-a-t-e-m-e-n-t-2-the-equation-a-x-2

slug:
st-a-t-e-m-e-n-t-1-if-the-general-equation-x-2-y-2-2-x-y-2-gx-2-f-y-4-0-represents-a-pair-of-real-lines-then-g-2-st-a-t-e-m-e-n-t-2-the-equation-a-x-2-66430

question:
<div class="question">$Statement- 1$: If the general equation $x^{2}+y^{2}+2 x y+2 g x+2 f y+4=0$ represents a pair of real lines then $|\mathrm{g}| \geq 2$. 
<br/> <br/>$Statement- 2$: The equation $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents pair
<br/> <br/>of real lines if $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Statement -1 is false, Statement- 2 is true</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Statement -1 is true, Statement- 2 is true; Statement -2 is a correct explanation for Statement-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Statement -1 is true, Statement- 2 is true; Statement -2 is not a correct explanation for Statement-1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Statement -1 is true, Statement- 2 is false</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Statement -1 is true, Statement- 2 is false</span> </div>

answer:
<div class="solution">The equation represents a pair of lines if $1 \cdot 1 \cdot 4+2 \cdot \mathrm{f} \cdot \mathrm{g} \cdot 1-1 \cdot \mathrm{f}^{2}-1 \cdot \mathrm{g}^{2}-4 \cdot 1^{2}=0$
<br/> <br/>$\Rightarrow(f-g)^{2}=0 \Rightarrow f=g$
<br/> <br/>The equation becomes $(x+y)^{2}+2 g(x+y)+4=0$ Which represents pair of parallel lines, which are real provided $(2 \mathrm{~g})^{2}-4 \cdot 4 \cdot 1 \geq 0 \Rightarrow|\mathrm{g}| \geq 2$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P2.db
title:
standard-entropy-of-x-2-y-2-and-x-y-2-are-6040-and-50-jk-1-mol-1-respectively-for-the-reaction-2-1-x-2-2-3-y-2-x-y-3-h-30-kj-to-be-at-equilibrium-the-

slug:
standard-entropy-of-x-2-y-2-and-x-y-2-are-6040-and-50-jk-1-mol-1-respectively-for-the-reaction-2-1-x-2-2-3-y-2-x-y-3-h-30-kj-to-be-at-equilibrium-the-72848

question:
<div class="question">Standard entropy of $X_2, Y_2$ and $X Y_2$ are 60,40 and $50 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}$, respectively. For the reaction, $\frac{1}{2} X_2+\frac{3}{2} Y_2 \longrightarrow X Y_3, \Delta H=-30 \mathrm{~kJ}$, to be at equilibrium, the temperature will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$1250 \mathrm{~K}$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$500 \mathrm{~K}$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$750 \mathrm{~K}$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$1000 \mathrm{~K}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$750 \mathrm{~K}$
<br/></span> </div>

answer:
<div class="solution">$\Delta S=S\left(X Y_3\right)-\frac{1}{2} S\left(X_2\right)-\frac{3}{2} S\left(Y_2\right)$
<br/>$=50-30-60=-40 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$
<br/>$\Delta H=-30 \mathrm{~kJ}=-3000 \mathrm{~J}$
<br/>$\Delta G=\Delta H-T \Delta S$
<br/>At equilibrium,
<br/>$\Delta G=0$
<br/>$T=\frac{\Delta H}{\Delta S}=\frac{-3000}{-40}=750 \mathrm{~K}$</div>


title: starch-can-be-used-as-an-indicator-for-the-detection-of-traces-of
slug: starch-can-be-used-as-an-indicator-for-the-detection-of-traces-of-25986
question: <div class="question">Starch can be used as an indicator for the detection of traces of</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2014']
answer: <div class="solution">$\mathrm{I}_{2}$ gives blue colour with starch.</div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">glucose in aqueous solution</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">proteins in blood</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">iodine in aqueous solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">urea in blood</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">iodine in aqueous solution</span> </div>


tags: ['Physics', 'Thermal Properties of Matter', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
steam-at-100-c-is-passed-into-1-1-kg-of-water-contained-in-a-calorimoter-of-water-equivalent-to-0-2-kg-at-15-c-till-the-temperature-of-the-calorimeter

slug:
steam-at-100-c-is-passed-into-1-1-kg-of-water-contained-in-a-calorimoter-of-water-equivalent-to-0-2-kg-at-15-c-till-the-temperature-of-the-calorimeter-32349

question:
<div class="question">Steam at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> is passed into <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>1</mn><mo> </mo><mi>kg</mi></math> of water contained in a calorimoter of water equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mi>kg</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> till the temperature of the calorimeter and its contents rises to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math>. The mass of steam condensed (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>kg</mi></math> ) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>130</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>065</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>260</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>135</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>130</mn></math></span> </div>

answer:
<div class="solution"><p>(a) According to principle of calorimetry,<br/>heat gained <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> heat lost<br/>Heat is lost by steam in two stages</p><p>(i) Change of state from steam at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> to water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mn>540</mn><mo>.</mo></math></p><p>(ii) To change water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> to water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo> </mo><mo>°</mo><mi mathvariant="normal">C</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mn>100</mn><mo>-</mo><mn>80</mn></mrow></mfenced><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is the mass of the steam condensed.</p><p>Total heat lost by steam is<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mn>540</mn><mo>+</mo><mi>m</mi><mo>×</mo><mn>20</mn><mo>=</mo><mi>m</mi><mo> </mo><mfenced><mrow><mn>540</mn><mo>+</mo><mn>20</mn></mrow></mfenced><mo>=</mo><mn>560</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>Heat gained by calorimeter and its contents is<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>.</mo><mn>1</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mn>80</mn><mo>-</mo><mn>15</mn></mrow></mfenced><mo>=</mo><mn>1</mn><mo>.</mo><mn>12</mn><mo>×</mo><mn>65</mn><mtext> cal </mtext></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mn>560</mn><mo> </mo><mi mathvariant="normal">m</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>12</mn><mo>×</mo><mn>65</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mi>m</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>12</mn><mo>×</mo><mn>65</mn></mrow><mn>560</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>130</mn><mo> </mo><mi>kg</mi></math></p></div>


tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
straight-lines-3-x-4-y-5-and-4-x-3-y-15-intersect-at-the-point-a-if-point-b-and-c-are-chosen-on-these-two-lines-such-that-a-b-a-c-then-the-possible-eq-1

slug:
straight-lines-3-x-4-y-5-and-4-x-3-y-15-intersect-at-the-point-a-if-point-b-and-c-are-chosen-on-these-two-lines-such-that-a-b-a-c-then-the-possible-eq-1-65844

question:
<div class="question">Straight lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>15</mn></math> intersect at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. If point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> are chosen on these two lines such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>C</mi></math>, then the possible equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> passing through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><p>The given straight lines are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>15</mn></math>. Clearly, these straight lines are perpendicular to each other <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> and intersect at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> are points on these lines such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math>.</p><p>From figure it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∠</mo><mi>B</mi><mo>=</mo><mo>∠</mo><mi>C</mi><mo>=</mo><mn>45</mn><mo>°</mo></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/84e5b535-e9a0-4211-ab75-00d5193aac21-d45.png" style="width: 310px; height: 200px;"/></p><p>Let slope of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>. Then,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mn>45</mn><mo>°</mo><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><mi>m</mi><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mrow><mn>1</mn><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>m</mi></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>±</mo><mn>1</mn><mo>=</mo><mfrac><mrow><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>4</mn><mo>-</mo><mn>3</mn><mi>m</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn><mo>=</mo><mo>±</mo><mfenced><mrow><mn>4</mn><mo>-</mo><mn>3</mn><mi>m</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>4</mn><mo>-</mo><mn>3</mn><mi>m</mi></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn><mo>=</mo><mo>-</mo><mn>4</mn><mo>+</mo><mn>3</mn><mi>m</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>7</mn></math></p><p>Hence, equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>7</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>7</mn><mi>y</mi><mo>-</mo><mn>14</mn><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>7</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>-</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></p><p>Hence, option (c) is correct.</p></div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
straight-lines-3-x-4-y-5-and-4-x-3-y-15-intersect-at-the-point-a-if-point-b-and-c-are-chosen-on-these-two-lines-such-that-a-b-a-c-then-the-possible-eq

slug:
straight-lines-3-x-4-y-5-and-4-x-3-y-15-intersect-at-the-point-a-if-point-b-and-c-are-chosen-on-these-two-lines-such-that-a-b-a-c-then-the-possible-eq-95934

question:
<div class="question">Straight lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>15</mn></math> intersect at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. If point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> are chosen on these two lines such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>C</mi></math>, then the possible equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> passing through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>7</mn><mo>=</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn><mtext> or </mtext><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><p>The given straight lines are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>15</mn></math>. Clearly, these straight lines are perpendicular to each other <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> and intersect at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>. Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> are points on these lines such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi><mi>C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math>.</p><p>From figure it is clear that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∠</mo><mi>B</mi><mo>=</mo><mo>∠</mo><mi>C</mi><mo>=</mo><mn>45</mn><mo>°</mo></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/84e5b535-e9a0-4211-ab75-00d5193aac21-d45.png" style="width: 310px; height: 200px;"/></p><p>Let slope of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>. Then,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mn>45</mn><mo>°</mo><mo>=</mo><mfenced close="|" open="|"><mfrac><mrow><mi>m</mi><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow><mrow><mn>1</mn><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>m</mi></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>±</mo><mn>1</mn><mo>=</mo><mfrac><mrow><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>4</mn><mo>-</mo><mn>3</mn><mi>m</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn><mo>=</mo><mo>±</mo><mfenced><mrow><mn>4</mn><mo>-</mo><mn>3</mn><mi>m</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn><mo>=</mo><mn>4</mn><mo>-</mo><mn>3</mn><mi>m</mi></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>m</mi><mo>+</mo><mn>3</mn><mo>=</mo><mo>-</mo><mn>4</mn><mo>+</mo><mn>3</mn><mi>m</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>7</mn></math></p><p>Hence, equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mi>C</mi></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>7</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>7</mn><mi>y</mi><mo>-</mo><mn>14</mn><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mo>-</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>7</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>-</mo><mn>7</mn><mi>y</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn></math></p><p>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>0</mn></math></p><p>Hence, option (c) is correct.</p></div>


title: strong-reducing-behaviour-of-h-3-po-2-is-due-to-1
slug: strong-reducing-behaviour-of-h-3-po-2-is-due-to-1-91745
question: <div class="question">Strong reducing behaviour of $\mathrm{H}_{3} \mathrm{PO}_{2}$ is due to</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Main']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">presence of one $-\mathrm{OH}$ group and two $\mathrm{P}-\mathrm{H}$ bonds</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">high electron gain enthalpy of phosphorus</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">high oxidation state of phosphorus</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">presence of two -OH groups and one $\mathrm{P}-\mathrm{H}$ bond</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">presence of one $-\mathrm{OH}$ group and two $\mathrm{P}-\mathrm{H}$ bonds</span> </div>

answer:
<div class="solution">The acids which contain $\mathrm{P}-\mathrm{H}$ bond have strong reducing properties. Thus $\mathrm{H}_{3} \mathrm{PO}_{2}$ acid is good reducing agent as it contains two P-H bonds. For example, it reduces $\mathrm{AgNO}_{3}$ to metallic silver.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9jdPCxj4Th9wFFql8EsIaNkzC-sXhad3i2tPe2Jn5yg.original.fullsize.png"/></div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
suppose-a-parabola-y-a-x-2-b-x-c-has-two-x-intercepts-one-positive-and-one-negative-and-its-vertex-is-2-2-then-which-of-the-following-is-true-1

slug:
suppose-a-parabola-y-a-x-2-b-x-c-has-two-x-intercepts-one-positive-and-one-negative-and-its-vertex-is-2-2-then-which-of-the-following-is-true-1-24718

question:
<div class="question">Suppose a parabola <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> has two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> intercepts, one positive and one negative, and its vertex is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>)</mo></math>, then which of the following is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mo>&gt;</mo><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> has two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercepts one is positive and one is negative and vertex is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>)</mo></math>. The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/2ea76fa3-d1ce-4ab3-ae57-cb22a3955241-image.png" style="width: 268px; height: 200px;"/></p><p>Clearly from graph</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi>b</mi><mi>a</mi></mfrac><mo>&gt;</mo><mn>0</mn><mo>⇒</mo><mo>-</mo><mi>b</mi><mo>&gt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>b</mi><mo>&lt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>&lt;</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mo>&lt;</mo><mn>0</mn><mo>,</mo><mi>a</mi><mi>c</mi><mo>&lt;</mo><mn>0</mn><mo>,</mo><mi>b</mi><mi>c</mi><mo>&gt;</mo><mn>0</mn></math></p></div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-1</span> </div>
source_db: MarksBatch2_P2.db
title:
suppose-p-q-r-0-and-system-of-equation-p-a-x-by-cz-0-a-x-q-b-y-cz-0-a-x-b-y-r-c-z-0-has-a-nontrivial-solution-then-value-of-p-a-q-b-r-c-is

slug:
suppose-p-q-r-0-and-system-of-equation-p-a-x-by-cz-0-a-x-q-b-y-cz-0-a-x-b-y-r-c-z-0-has-a-nontrivial-solution-then-value-of-p-a-q-b-r-c-is-61134

question:
<div class="question">Suppose $\mathrm{p}, \mathrm{q}, \mathrm{r} \neq 0$ and system of equation $(\mathrm{p}+\mathrm{a}) \mathrm{x}+\mathrm{by}+\mathrm{cz}=0$
<br/> <br/>$a x+(q+b) y+c z=0$
<br/> <br/>$a x+b y+(r+c) z=0$ has a non-trivial solution,
<br/> <br/>then value of $\frac{a}{p}+\frac{b}{q}+\frac{c}{r}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">-1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Since the system is homogeneous and has a non-trivial solution
<br/> <br/>$0=\left|\begin{array}{ccc}p+a &amp; b &amp; c \\ a &amp; q+b &amp; c \\ a &amp; b &amp; r+c\end{array}\right|=\left|\begin{array}{ccc}p &amp; -q &amp; 0 \\ 0 &amp; q &amp; -r \\ a &amp; b &amp; r+c\end{array}\right|$
<br/> <br/>using $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}$ and $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$
<br/> <br/>Expanding along $\mathrm{C}$, we get $\Rightarrow \mathrm{p}[\mathrm{q}(\mathrm{r}+\mathrm{c})+\mathrm{br}]+\mathrm{aqr}=0$
<br/> <br/>$\Rightarrow \mathrm{pqr}+\mathrm{pqc}+\mathrm{prb}+\mathrm{qra}=0$
<br/> <br/>$\Rightarrow \frac{a}{p}+\frac{b}{q}+\frac{c}{r}=-1$</div>


title: suppose-that-f-0-3-and-f-x-5-for-all-values-of-x-then-the-largest-value-which-f-2-can-attain-is
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P2.db
slug:
suppose-that-f-0-3-and-f-x-5-for-all-values-of-x-then-the-largest-value-which-f-2-can-attain-is-54958

question:
<div class="question">Suppose that $f(0)=-3$ and $f^{\prime}(x) \leq 5$ for all values of $x$ Then, the largest value which $f(2)$ can attain is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9</span> </li> </ul>

answer:
<div class="solution">Using LMVT in [0,2] $\frac{f(2)-f(0)}{2-0}=f^{\prime}(c),$ where $c \in(0,2)$
<br/> <br/>$\frac{f(2)+3}{2} \leq 5 \Rightarrow f(2) \leq 7$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
title:
suppose-the-gravitational-force-varies-inversely-as-the-nth-power-of-distance-then-the-time-period-of-a-planet-in-circular-orbit-of-radius-r-around-th

slug:
suppose-the-gravitational-force-varies-inversely-as-the-nth-power-of-distance-then-the-time-period-of-a-planet-in-circular-orbit-of-radius-r-around-th-97554

question:
<div class="question">Suppose the gravitational force varies inversely as the nth power of distance. Then the time period of a planet in circular orbit of radius ' $\mathrm{R}$ ' around the sun will be proportional to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{R}^{\mathrm{n}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{R}^{\left(\frac{\mathrm{n}-1}{2}\right)}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{R}\left(\frac{\mathrm{n}+1}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{R}^{\left(\frac{\mathrm{n}-2}{2}\right)}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{R}\left(\frac{\mathrm{n}+1}{2}\right)$</span> </div>

answer:
<div class="solution">$$<br/>\mathrm{F}=\mathrm{KR}^{-\mathrm{n}}=\mathrm{MR} \omega^{2} \Rightarrow \omega^{2}=\mathrm{KR}^{-(\mathrm{n}+1)}<br/>$$<br/>or $\omega=\mathrm{K}^{\prime} \mathrm{R}^{\frac{-(\mathrm{n}+1)}{2}}$<br/>[where $\mathrm{K}^{\prime}=\mathrm{K}^{1 / 2}$, a constant $]$<br/>$\frac{2 \pi}{T} \alpha R^{\frac{-(n+1)}{2}} \quad \therefore T \alpha R^{\frac{(n+1)}{2}}$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
suppose-the-law-of-gravitational-attraction-suddenly-changes-and-becomes-an-inverse-cube-law-ie-f-r-3-1-but-still-remaining-a-central-force-then

slug:
suppose-the-law-of-gravitational-attraction-suddenly-changes-and-becomes-an-inverse-cube-law-ie-f-r-3-1-but-still-remaining-a-central-force-then-66352

question:
<div class="question">Suppose the law of gravitational attraction suddenly changes and becomes an inverse cube law i.e. F $\propto \frac{1}{r^{3}}$, but still remaining a central force. Then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Kepler's law of area still holds</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Kepler's law of period still holds</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Kepler's law of area and period still holds</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">neither the law of area, nor the law of period still holds</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Kepler's law of area still holds</span> </div>

answer:
<div class="solution">According to kepler's law of area $\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=\frac{\mathrm{L}}{2 \mathrm{~m}}$
<br/> <br/>For central forces, torque $=0$
<br/> <br/>$\therefore \mathrm{L}=$ constant
<br/> <br/>$\therefore \frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=$ constant</div>


title: suppose-the-limit-l-lim-n-n-0-1-1-1-x-2-n-d-x-exists-and-is-larger-than-1-2-then-1
slug: suppose-the-limit-l-lim-n-n-0-1-1-1-x-2-n-d-x-exists-and-is-larger-than-1-2-then-1-17122
tags: ['Mathematics', 'Definite Integration', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">Suppose the limit <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><msqrt><mi>n</mi></msqrt><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mn>1</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></mfrac><mi>d</mi><mi>x</mi></math> exists and is larger than <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>&lt;</mo><mi>L</mi><mo>&lt;</mo><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>&lt;</mo><mi>L</mi><mo>&lt;</mo><mn>3</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&lt;</mo><mi>L</mi><mo>&lt;</mo><mn>4</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>≥</mo><mn>4</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>&lt;</mo><mi>L</mi><mo>&lt;</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><msqrt><mi>n</mi></msqrt><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></mfrac></math></p><p>We know that, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup><mo>&gt;</mo><mn>1</mn><mo>+</mo><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mn>1</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></mfrac><mo>&lt;</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></mfrac><mo>&lt;</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mn>1</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></mfrac><mo>&lt;</mo><mfrac><mn>1</mn><msqrt><mi>n</mi></msqrt></mfrac><msubsup><mfenced close="]" open="["><mrow><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msqrt><mi>n</mi></msqrt><mi>x</mi></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><msqrt><mi>n</mi></msqrt></mfrac><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msqrt><mi>n</mi></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><msqrt><mi>n</mi></msqrt><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mn>1</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></mfrac><mi>d</mi><mi>x</mi><mo>&lt;</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><msqrt><mi>n</mi></msqrt><mfrac><mn>1</mn><msqrt><mi>n</mi></msqrt></mfrac><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msqrt><mi>n</mi></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>L</mi><mo>=</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></p></div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
title:
taking-rydbergs-constant-r-h-1097-1-0-7-m-first-and-second-wavelength-of-balmer-series-in-hydrogen-spectrum-is

slug:
taking-rydbergs-constant-r-h-1097-1-0-7-m-first-and-second-wavelength-of-balmer-series-in-hydrogen-spectrum-is-35407

question:
<div class="question">Taking Rydberg's constant $\mathrm{R}_{\mathrm{H}}=1.097 \times 10^{7} \mathrm{~m}$, first and second wavelength of Balmer series in hydrogen spectrum is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2000 \mathrm{~A}, 3000 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1575 \mathrm{~A}, 2960 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$6529 \mathrm{~A}, 4280 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$6552 \mathrm{~A}, 4863 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6552 \mathrm{~A}, 4863 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{\mathrm{n}_{1}^{2}}-\frac{1}{\mathrm{n}_{2}^{2}}\right]$. For first wavelength, $\mathrm{n}_{1}$<br/>$$<br/>=2, \mathrm{n}_{2}=3<br/>$$<br/>$\Rightarrow \lambda_{1}=6563 \AA$. For second wavelength, $\mathrm{n}_{1}$<br/>$=2, \mathrm{n}_{2}=4 \Rightarrow \lambda_{2}=4861 \AA$</div>


title: tan-1-4-1-tan-1-9-2-equal-to
slug: tan-1-4-1-tan-1-9-2-equal-to-95425
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">$\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)$ equal to-</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \cos ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2} \sin ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2} \tan ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\tan ^{-1}\left(\frac{1}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1}\left(\frac{1}{2}\right)$</span> </div>

answer:
<div class="solution">$\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)$
<br/>$\quad=\tan ^{-1}\left[\frac{\frac{1}{4}+\frac{2}{9}}{1-\frac{1}{4} \times \frac{2}{9}}\right]=\tan ^{-1}\left[\frac{1}{2}\right]$</div>


title: tan-1-4-1-tan-1-9-2-is-equal-to
slug: tan-1-4-1-tan-1-9-2-is-equal-to-11900
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">$\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2} \cos ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{2} \sin ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2} \tan ^{-1}\left(\frac{3}{5}\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\tan ^{-1}\left(\frac{1}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1}\left(\frac{1}{2}\right)$</span> </div>

answer:
<div class="solution">$\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{2}{9}\right)=\tan ^{-1}\left\{\frac{\frac{1}{4}+\frac{2}{9}}{1-\frac{1}{4} \times \frac{2}{9}}\right\}$<br/>$=\tan ^{-1}\left\{\frac{9+8}{36-2}\right\}=\tan ^{-1}\left(\frac{1}{2}\right)$</div>


title: tan-1-x-dx-is-equal-to
slug: tan-1-x-dx-is-equal-to-62827
question: <div class="question">$\int \tan ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}$ is equal to</div>
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(x+1) \tan ^{-1} \sqrt{x}-\sqrt{x}+C$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x \tan ^{-1} \sqrt{x}-\sqrt{x}+C$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{x}-x \tan ^{-1} \sqrt{x}+C$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{x}-(x+1) \tan ^{-1} \sqrt{x}+C$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(x+1) \tan ^{-1} \sqrt{x}-\sqrt{x}+C$</span> </div>

answer:
<div class="solution">We have, $I=\int 1 \cdot \tan ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}$<br/>Using by parts,<br/>$\begin{aligned}<br/>I &amp; =\tan ^{-1} \sqrt{x} \cdot(x)-\int \frac{1}{1+x} \times \frac{1}{2 \sqrt{x}} \times x d x \\<br/>&amp; =x \tan ^{-1} \sqrt{x} \cdot-\int \frac{x}{(1+x) 2 \sqrt{x}} d x \\<br/>&amp; =x \tan ^{-1} \sqrt{x}-\int\left(\frac{1+x}{(1+x) 2 \sqrt{x}}-\frac{1}{(1+x) 2 \sqrt{x}}\right) d x \\<br/>&amp; =x \tan ^{-1} \sqrt{x}-\int \frac{d x}{2 \sqrt{x}}+\int \frac{d x}{2 \sqrt{x}(1+x)} \\<br/>&amp; =x \tan ^{-1} \sqrt{x}-\sqrt{x}+\tan ^{-1} \sqrt{x}+C \\<br/>&amp; =(x+1) \tan ^{-1} \sqrt{x}-\sqrt{x}+C<br/>\end{aligned}$</div>


title: tan-20circtan-40circsqrt3-tan-20circ-tan-40circ-is-equal-to
slug: tan-20circtan-40circsqrt3-tan-20circ-tan-40circ-is-equal-to-89455
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
question:
<div class="question">\(\tan 20^{\circ}+\tan 40^{\circ}+\sqrt{3} \tan 20^{\circ} \tan 40^{\circ}\) is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\sqrt{3} / 2\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\sqrt{3} / 4\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\sqrt{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\sqrt{3}\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>\sqrt{3} &amp; =\tan 60^{\circ}=\tan \left(40^{\circ}+20^{\circ}\right) \\<br/>&amp; =\frac{\tan 40^{\circ}+\tan 20^{\circ}}{1-\tan 40^{\circ} \tan 20^{\circ}} \\<br/>\therefore \quad &amp; \sqrt{3}-\sqrt{3} \tan 40^{\circ} \tan 20^{\circ} \\<br/>&amp; =\tan 40^{\circ}+\tan 20^{\circ}<br/>\end{aligned}\)<br/>Hence \(\tan 40^{\circ}+\tan 20^{\circ}+\sqrt{3} \tan 40^{\circ} \tan 20^{\circ}\) \(=\sqrt{3}\)</div>


title: tangents-are-drawn-from-the-origin-to-the-curve-y-cos-x-their-points-of-contact-lie-on
slug: tangents-are-drawn-from-the-origin-to-the-curve-y-cos-x-their-points-of-contact-lie-on-76936
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
question:
<div class="question">Tangents are drawn from the origin to the curve $\mathrm{y}=\cos \mathrm{x} .$ Their points of contact lie on</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2} y^{2}=y^{2}-x^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x^{2} y^{2}=x^{2}+y^{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x^{2} y^{2}=x^{2}-y^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x^{2} y^{2}=x^{2}-y^{2}$</span> </div>

answer:
<div class="solution">Let $\left(x_{1}, y_{1}\right)$ be one of the points of contact. Given curve is $y=\cos x$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=-\sin \mathrm{x} \\
<br/> <br/>\left.\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}\right|_{\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)}=-\sin \mathrm{x}_{1}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Now the equation of the tangent at $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ is
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>&amp; y-y_{1}\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}\left(x-x_{1}\right) \\
<br/> <br/>\Rightarrow &amp; y-y_{1}=-\sin x_{1}\left(0-x_{1}\right)
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Since, it is given that equation of tangent passes through origin.
<br/> <br/>$$
<br/> <br/>\begin{array}{lr}
<br/> <br/>\therefore &amp; 0-y_{1}=-\sin x_{1}\left(0-x_{1}\right) \\
<br/> <br/>\Rightarrow &amp; y_{1}=-x_{1} \sin x_{1}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Also, point $\left(x_{1}, y_{1}\right)$ lies on $y=\cos x$
<br/> <br/>$$
<br/> <br/>\therefore \quad y_{1}=\cos x_{1}
<br/> <br/>$$
<br/> <br/>From Eqs. (i), (ii), we get
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>&amp; \sin ^{2} x_{1}+\cos ^{2} x_{1}=\frac{y_{1}^{2}}{x_{1}^{2}}+y_{1}^{2}=1 \\
<br/> <br/>\Rightarrow &amp; x_{1}^{2}=y_{1}^{2}+y_{1}^{2} x_{1}^{2}
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Hence, the locus of $\left(x_{1}, y_{1}\right)$ is
<br/> <br/>$$
<br/> <br/>x^{2}=y^{2}+y^{2} x^{2} \Rightarrow x^{2} y^{2}=x^{2}-y^{2}
<br/> <br/>$$</div>


title: the-100text-th-term-of-the-sequence-12233344-44-ldots-is
slug: the-100text-th-term-of-the-sequence-12233344-44-ldots-is-48235
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">14</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The \(100^{\text {th }}\) term of the sequence \(1,2,2,3,3,3,4,4\), \(4,4, \ldots\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">13</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">14</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">15</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; 1^{\text {st }} \text { term } \rightarrow 1,2^{\text {nd }} \text { term }=2,4^{\text {th }} \text { term } \rightarrow 3, \\<br/>&amp; 7^{\text {th }} \text { term } \rightarrow 4,11^{\text {th }} \text { term } \rightarrow 5, \ldots<br/>\end{aligned}\)<br/>Series is \(1,2,4,7,11, \ldots\)<br/>\(a_n=1+\frac{n(n-1)}{2}=\frac{n^2-n+2}{2}\)<br/>If \(n=14\), then \(a_n=92, \mathrm{If} n=15\), then \(a_n=106\).</div>


tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
title:
the-activation-energy-for-a-simple-chemical-reaction-a-b-is-e-a-in-forward-direction-the-activation-energy-for-reverse-reaction

slug:
the-activation-energy-for-a-simple-chemical-reaction-a-b-is-e-a-in-forward-direction-the-activation-energy-for-reverse-reaction-68209

question:
<div class="question">The activation energy for a simple chemical reaction $\mathrm{A} \rightarrow \mathrm{B}$ is $E_{a}$ in forward direction. The activation energy for reverse reaction</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">is always less than $E_{a}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">can be less than or more than $E_{a}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">is always double of $E_{a}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">is negative of $E_{a}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">can be less than or more than $E_{a}$</span> </div>

answer:
<div class="solution">Since the nature of reaction (i.e. exothermic or endothermic) is not given, $\mathrm{E}_{\mathrm{a}}$ for reverse reaction can be more or less.</div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
title:
the-activity-of-a-radioactive-sample-is-measured-as-n-0-counts-per-minute-at-t-0-and-n-0-e-counts-per-minute-at-t-5-min-the-time-in-minutes-at-which-t-1

slug:
the-activity-of-a-radioactive-sample-is-measured-as-n-0-counts-per-minute-at-t-0-and-n-0-e-counts-per-minute-at-t-5-min-the-time-in-minutes-at-which-t-1-25574

question:
<div class="question">The activity of a radioactive sample is measured as <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>N</mi><mn>0</mn></msub></math> counts per minute at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>N</mi><mn>0</mn></msub><mi>e</mi></mfrac></math> counts per minute at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>5</mn><mo> </mo><mi>min</mi></math>. The time (in minutes) at which the activity reduces to half its value is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mi>e</mi></msub><mfrac><mn>2</mn><mn>5</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mrow><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msub><mi>log</mi><mn>10</mn></msub><mn>2</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Fraction remains after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> half-lives </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><msub><mi>N</mi><mn>0</mn></msub></mfrac><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mfrac><mi>t</mi><mi>T</mi></mfrac></msup></math></p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><msub><mi>N</mi><mn>0</mn></msub><mi>e</mi></mfrac><mo>⇒</mo><mfrac><msub><mi>N</mi><mn>0</mn></msub><mrow><mi>e</mi><msub><mi>N</mi><mn>0</mn></msub></mrow></mfrac><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mfrac><mn>5</mn><mi>T</mi></mfrac></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mn>1</mn><mi>e</mi></mfrac><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mfrac><mn>5</mn><mi>T</mi></mfrac></msup></math></p><p>Taking log on both sides, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mn>1</mn><mo>-</mo><mi>log</mi><mi>e</mi><mo>=</mo><mfrac><mn>5</mn><mi>T</mi></mfrac><mi>log</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mn>5</mn><mi>T</mi></mfrac><mfenced><mrow><mo>-</mo><mi>log</mi><mn>2</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>T</mi><mo>=</mo><mn>5</mn><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></p><p>Now, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> be the time after which activity reduces to half</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mo>=</mo><msup><mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></mfenced><mrow><mi>t</mi><mo>/</mo><mn>5</mn><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>t</mi><mo>'</mo></msup><mo>=</mo><mn>5</mn><msub><mi>log</mi><mi>e</mi></msub><mn>2</mn></math></p></div>


title: the-addition-of-hbr-to-1-butene-gives-a-mixture-of-products-x-y-and-z-the-mixture-consists-of-1
tags: ['Chemistry', 'General Organic Chemistry', 'JEE Main']
source_db: MarksBatch2_P2.db
slug:
the-addition-of-hbr-to-1-butene-gives-a-mixture-of-products-x-y-and-z-the-mixture-consists-of-1-17854

question:
<div class="question"><p>The addition of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HBr</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>-butene gives a mixture of products <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">z</mi></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/20a136e8-0593-4984-a8df-0777e183bb31-image.png" style="width: 487px; height: 162px;"/></p><p>The mixture consists of</p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math> as major and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">z</mi></math> as minor products</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math> as major, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">z</mi></math> as minor products</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math> as minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">z</mi></math> as major products</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math> as minor and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">z</mi></math> as major products</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math> as major and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">z</mi></math> as minor products</span> </div>

answer:
<div class="solution"><p>The alkene is unsymmetrical, hence will follow Markownikoff's rule to give major product.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c11f11a6-5587-47c0-b37e-e6797e116bff-image.png" style="width: 411px; height: 184px;"/></p><p>since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math> contains, a chiral carbon, it exists in two enantiomers (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">y</mi></math>) which are mirror images of each other.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/1660552e-1626-4b50-9ed6-8dd960f85287-image.png" style="width: 341px; height: 195px;"/></p></div>


title: the-alcohol-products-of-the-reduction-of-2-methyl3pentanone-with-lialh-4-is-are-1
slug: the-alcohol-products-of-the-reduction-of-2-methyl3pentanone-with-lialh-4-is-are-1-42664
tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">The alcohol product(s) of the reduction of 2 -methyl-3-pentanone with $\mathrm{LiAlH}_{4}$ is (are)
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/FL7qJ-fgwoOJ81zrSENQyzaDP9qqepSel4wRRaqqLzQ.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a single enantiomer</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">racemic mixture</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">two diastereoisomers</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">two structural isomers</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">racemic mixture</span> </div>

answer:
<div class="solution">The resultingcompound is $\mathrm{C}_{3} \mathrm{H}_{7} \mathrm{CH}(\mathrm{OH}) \mathrm{C}_{2} \mathrm{H}_{5}$
<br/> <br/>which is optically inactive and reaction leads the racemisation</div>


tags: ['Chemistry', 's Block Elements', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
title:
the-alkali-metals-form-saltlike-hydrides-by-the-direct-synthesis-at-elevated-temperature-the-thermal-stability-of-these-hydrides-decreases-in-which-of-1

slug:
the-alkali-metals-form-saltlike-hydrides-by-the-direct-synthesis-at-elevated-temperature-the-thermal-stability-of-these-hydrides-decreases-in-which-of-1-16995

question:
<div class="question">The alkali metals form salt-like hydrides by the direct synthesis at elevated temperature. The thermal stability of these hydrides decreases in which of the following orders?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CsH}&gt;\mathrm{RbH}&gt;\mathrm{KH}&gt;\mathrm{NaH}&gt;\mathrm{LiH}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{KH}&gt;\mathrm{NaH}&gt;\mathrm{LiH}&gt;\mathrm{CsH}&gt;\mathrm{RbH}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{NaH}&gt;\mathrm{LiH}&gt;\mathrm{KH}&gt;\mathrm{RbH}&gt;\mathrm{CsH}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{LiH}&gt;\mathrm{NaH}&gt;\mathrm{KH}&gt;\mathrm{RbH}&gt;\mathrm{CsH}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{LiH}&gt;\mathrm{NaH}&gt;\mathrm{KH}&gt;\mathrm{RbH}&gt;\mathrm{CsH}$</span> </div>

answer:
<div class="solution">The stability of alkali metal hydrides decreases from Li to Cs. It is due to the fact that $\mathrm{M}-\mathrm{H}$ bonds becomes weaker with increase in size of alkali metals as we move down the group from $\mathrm{Li}$ to $\mathrm{Cs}$. Thus the order of stability of hydrides is $\mathrm{LiH}&gt;\mathrm{NaH}&gt;\mathrm{KH}&gt;\mathrm{RbH}&gt;\mathrm{CsH}$</div>


title: the-amount-of-h-2-o-2-present-in-1-litre-of-15-n-h-2-o-2-solution-is
slug: the-amount-of-h-2-o-2-present-in-1-litre-of-15-n-h-2-o-2-solution-is-43460
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
question:
<div class="question">The amount of $\mathrm{H}_{2} \mathrm{O}_{2}$ present in 1 litre of $1.5 \mathrm{~N} \mathrm{H}_{2} \mathrm{O}_{2}$ solution, is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$25.5 \mathrm{~g}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.0 \mathrm{~g}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8.0 \mathrm{~g}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.5 \mathrm{~g}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$25.5 \mathrm{~g}$</span> </div>

answer:
<div class="solution">Molecular weight of $\mathrm{H}_{2} \mathrm{O}_{2}=34$ Equivalent weight of $\mathrm{H}_{2} \mathrm{O}_{2}=17$
<br/> <br/>$\therefore 1 \mathrm{~L}$ of $1 \mathrm{~N} \mathrm{H}_{2} \mathrm{O}_{2} \mathrm{has}=17 \mathrm{~g}$ of $\mathrm{H}_{2} \mathrm{O}$
<br/> <br/>$\therefore 1$ Lof $1.5 \mathrm{NH}_{2} \mathrm{O}_{2}$ has $=1.5 \times 17=25.5 \mathrm{~g} \mathrm{of} \mathrm{H}_{2} \mathrm{O}_{2}$</div>


title: the-amplitude-of-sin-5-i-1-cos-5
slug: the-amplitude-of-sin-5-i-1-cos-5-96845
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">The amplitude of $\sin \frac{\pi}{5}+i\left(1-\cos \frac{\pi}{5}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi / 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \pi / 5$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\pi / 10$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi / 15$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\pi / 10$</span> </div>

answer:
<div class="solution">$\sin \frac{\pi}{5}+\mathrm{i}\left(1-\cos \frac{\pi}{5}\right)$
<br/>$=2 \sin \frac{\pi}{10} \cos \frac{\pi}{10}+\mathrm{i} 2 \sin ^{2} \frac{\pi}{10}$
<br/>$=2 \sin \frac{\pi}{10}\left(\cos \frac{\pi}{10}+\mathrm{i} \sin \frac{\pi}{10}\right)$
<br/>For amplitude, $\tan \theta=\frac{\pi}{\sin \frac{\pi}{10}}{\cos \frac{\pi}{10}}=\tan \frac{\pi}{10}$
<br/>$\Rightarrow \theta=\frac{\pi}{10}$</div>


title: the-angle-between-any-two-diagonal-of-a-cube-is-1
slug: the-angle-between-any-two-diagonal-of-a-cube-is-1-73299
question: <div class="question">The angle between any two diagonal of a cube is</div>
tags: ['Mathematics', 'Vector Algebra', 'JEE Main']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$45^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$60^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\tan ^{-1}(2 \sqrt{2})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan ^{-1}(2 \sqrt{2})$</span> </div>

answer:
<div class="solution">for a unit cube unit vector along the diagonal
<br/> <br/>$$
<br/> <br/>\mathrm{OP}=\frac{1}{\sqrt{3}}(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})
<br/> <br/>$$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/7NTWetRK4yxOBniSvpXeh_Q4dVkdpauFDXCZLGDxbqY.original.fullsize.png"/><br/> <br/> <br/>unit vector along the diagonal
<br/> <br/>$\mathrm{CD}=\frac{1}{\sqrt{3}}(\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}})$
<br/> <br/>$\therefore \cos \theta=\frac{1}{3}(1+1-1)=\frac{1}{3} \quad \therefore \tan \theta=2 \sqrt{2}$</div>


title: the-angle-between-the-lines-whose-direction-cosine-satisfy-the-equations-l-m-n-0-and-l-2-m-2-n-2-is
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
slug:
the-angle-between-the-lines-whose-direction-cosine-satisfy-the-equations-l-m-n-0-and-l-2-m-2-n-2-is-81223

question:
<div class="question">The angle between the lines whose direction cosine satisfy the equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>+</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>l</mi><mn>2</mn></msup><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>3</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>6</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>3</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>+</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>l</mi><mn>2</mn></msup><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>ii</mi><mo>)</mo></math></p><p>Putting the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> in Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ii</mi><mo>)</mo></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>m</mi><mo>+</mo><mi>n</mi><msup><mo>)</mo><mn>2</mn></msup><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>m</mi><mi>n</mi><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>m</mi><mi>n</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>0</mn></math></p><p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>l</mi><mo>=</mo><mo>-</mo><mi>n</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Direction cosines are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>,</mo><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math>.</p><p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>l</mi><mo>=</mo><mo>-</mo><mi>m</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Direction cosines are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>,</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>,</mo><mn>0</mn></mrow></mfenced></math>.</p><p>Angle between lines are </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>θ</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><msqrt><mn>2</mn></msqrt></mfrac><mo>×</mo><mo>-</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>0</mn><mo>×</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced><mo>+</mo><mfenced><mrow><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>×</mo><mn>0</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mi>θ</mi><mo>=</mo><mi>π</mi><mo>/</mo><mn>3</mn></math></p></div>


tags: ['Physics', 'Magnetic Properties of Matter', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-angle-of-dip-at-a-place-is-3-7-and-the-vertical-component-of-the-earths-magnetic-field-is-6-1-0-5-t-the-earths-magnetic-field-at-this-place-is-tan

slug:
the-angle-of-dip-at-a-place-is-3-7-and-the-vertical-component-of-the-earths-magnetic-field-is-6-1-0-5-t-the-earths-magnetic-field-at-this-place-is-tan-12895

question:
<div class="question">The angle of dip at a place is $37^{\circ}$ and the vertical component of the earth's magnetic field is $6 \times 10^{-5} \mathrm{~T}$. The earth's magnetic field at this place is $\left(\tan 37^{\circ}=\frac{3}{4}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7 \times 10^{-5} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$6 \times 10^{-5} \mathrm{~T}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \times 10^{-5} \mathrm{~T}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$10^{-4} \mathrm{~T}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10^{-4} \mathrm{~T}$</span> </div>

answer:
<div class="solution">$\tan \theta=\frac{\mathrm{V}}{\mathrm{H}}=\frac{3}{4}\left[\because \tan 37^{\circ}=\frac{3}{4}\right]$
<br/> <br/>$\therefore \mathrm{V}=\frac{3}{4} \mathrm{H}$
<br/> <br/>$\mathrm{V}=6 \times 10^{-5} \mathrm{~T}$
<br/> <br/>$\mathrm{H}=\frac{4}{3} \times 6 \times 10^{-5} \mathrm{~T}=8 \times 10^{-5} \mathrm{~T}$
<br/> <br/>$\therefore \mathrm{B}_{\text {total }}=\sqrt{\mathrm{V}^{2}+\mathrm{H}^{2}}=\sqrt{(36+64)} \times 10^{-5}$
<br/> <br/>$=10 \times 10^{-5}=10^{-4} \mathrm{~T} .$</div>


title: the-angle-of-intersection-of-the-two-circles-x-2-y-2-2-x-2-y-0-and-x-2-y-2-4-is
slug: the-angle-of-intersection-of-the-two-circles-x-2-y-2-2-x-2-y-0-and-x-2-y-2-4-is-57892
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The angle of intersection of the two circles $x^{2}+y^{2}-2 x-2 y=0$ and $x^{2}+y^{2}=4,$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$60^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$90^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$45^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$45^{\circ}$</span> </div>

answer:
<div class="solution">Here circles are
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>x^{2}+y^{2}-2 x-2 y=0...(1) \\
<br/> <br/>x^{2}+y^{2}=4...(2) \\
<br/> <br/>\text { Now, } \mathrm{C}_{1}(1,1), r_{1}=\sqrt{1^{2}+1^{2}}=\sqrt{2} \\
<br/> <br/>C_{2}(0,0), r_{2}=2
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>If $\theta$ is the angle of intersection then
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\cos \theta=\frac{r_{1}^{2}+r_{2}^{2}-\left(c_{1} c_{2}\right)^{2}}{2 r_{1} r_{2}} \\
<br/> <br/>=\frac{2+4-(\sqrt{2})^{2}}{2 \cdot \sqrt{2.2}}=\frac{1}{\sqrt{2}} \Rightarrow \theta=45^{\circ}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>L</span> </div>
source_db: MarksBatch2_P2.db
title:
the-angular-momentum-for-the-electron-in-bohrs-orbit-is-l-if-the-electron-is-assumed-to-revolve-in-second-orbit-of-hydrogen-atom-then-the-change-in-an-1

slug:
the-angular-momentum-for-the-electron-in-bohrs-orbit-is-l-if-the-electron-is-assumed-to-revolve-in-second-orbit-of-hydrogen-atom-then-the-change-in-an-1-41565

question:
<div class="question">The angular momentum for the electron in Bohr's orbit is L. If the electron is assumed to revolve in second orbit of hydrogen atom, then the change in angular momentum will be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{L}}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>zero</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>L</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>2L</span> </li> </ul>

answer:
<div class="solution">From the Bohr's quantization rule<br/>Angular momentum $\mathrm{L}=\frac{\mathrm{nh}}{2 \pi}, \mathrm{L}_1=\frac{1 . \mathrm{h}}{2 \pi}=\mathrm{L}$ ( $n=1$ for first orbit)<br/>in ( $n=2$ for second orbit) $L_2=\frac{2 h}{2 \pi}=2 L$<br/>Hence, change in angular momentum $\mathrm{L}_2-\mathrm{L}_1=2 \mathrm{~L}-$ $\mathrm{L}=\mathrm{L}$</div>


title: the-angular-size-of-the-central-maxima-due-to-a-single-slit-diffraction-is-a-slit-width
slug: the-angular-size-of-the-central-maxima-due-to-a-single-slit-diffraction-is-a-slit-width-97251
tags: ['Physics', 'Wave Optics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The angular size of the central maxima due to a single slit diffraction is $\mathrm{a} \rightarrow$ slit width $)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\lambda}{\mathrm{a}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{2 \lambda}{\text { a }}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3 \lambda}{2 \mathrm{a}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\lambda}{2 \mathrm{a}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \lambda}{\text { a }}$</span> </div>

answer:
<div class="solution">Angular size of central maxima is
<br/> <br/>$$
<br/> <br/>2 \times\left(\frac{\lambda}{a}\right)=\frac{2 \lambda}{a}
<br/> <br/>$$</div>


title: the-anthracene-is-purified-by
slug: the-anthracene-is-purified-by-93623
question: <div class="question">The anthracene is purified by</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">crystallisation</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">filtration</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">distillation</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">sublimation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">sublimation</span> </div>

answer:
<div class="solution">Anthracene is purified by sublimation. In sublimation, a solid is converted directly into gaseous state on heating without passing through liquid phase.</div>


tags: ['Physics', 'Mechanical Properties of Fluids', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
title:
the-approximate-depth-of-an-ocean-is-2700-m-the-compressibility-of-water-is-454-1-0-11-pa-1-and-density-of-water-is-1-0-3-kg-m-3-what-fractional-compr

slug:
the-approximate-depth-of-an-ocean-is-2700-m-the-compressibility-of-water-is-454-1-0-11-pa-1-and-density-of-water-is-1-0-3-kg-m-3-what-fractional-compr-97515

question:
<div class="question">The approximate depth of an ocean is $2700 \mathrm{~m}$. The compressibility of water is $45.4 \times 10^{-11} \mathrm{~Pa}^{-1}$ and density of water is $10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. What fractional compression of water will be obtained at the bottom of the ocean?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.0 \times 10^{-2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.2 \times 10^{-2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.4 \times 10^{-2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.8 \times 10^{-2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.2 \times 10^{-2}$</span> </div>

answer:
<div class="solution">Compressibility of water, $\mathrm{K}=45.4 \times 10^{-11} \mathrm{~Pa}^{-1}$
<br/> <br/>density of water $\mathrm{P}=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$
<br/> <br/>depth of ocean, $\mathrm{h}=2700 \mathrm{~m}$ We have to find $\frac{\Delta \mathrm{V}}{\mathrm{V}}=?$
<br/> <br/>As we know, compressibility, $\mathrm{K}=\frac{1}{\mathrm{~B}}=\frac{(\Delta \mathrm{V} / \mathrm{V})}{\mathrm{P}}(\mathrm{P}=\mathrm{Pgh})$
<br/> <br/>So, $(\Delta \mathrm{V} / \mathrm{V})=\mathrm{KPgh}$
<br/> <br/>$=45.4 \times 10^{-11} \times 10^{3} \times 10 \times 2700$
<br/> <br/>$=1.2258 \times 10^{-2}$</div>


title: the-area-bounded-by-curves-x-1-2-y-2-1-and-x-2-y-2-1-is-1
slug: the-area-bounded-by-curves-x-1-2-y-2-1-and-x-2-y-2-1-is-1-47310
question: <div class="question">The area bounded by curves $(x-1)^{2}+y^{2}=1$ and $x^{2}+y^{2}=1$ is</div>
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\frac{2 \pi}{3}-\frac{\sqrt{3}}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2 \pi}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{3}}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2 \pi}{3}+\frac{\sqrt{3}}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{2 \pi}{3}-\frac{\sqrt{3}}{2}\right)$</span> </div>

answer:
<div class="solution">Given circles are $x^{2}+y^{2}=1 \ldots$ (i) and $(x-1)^{2}+y^{2}=1 \ldots$ (ii)
<br/> <br/>Centre of (i) is $\mathrm{O}(0,0)$ and radius $=1$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/h-abKd3JCjcK1BeOZKRiAZ6yaZvRisMxeLCXMht6Nb0.original.fullsize.png"/><br/> <br/> <br/>Both these circle are symmetrical about $x$-axis solving (i) and (ii), we get, $-2 x+1=0$
<br/> <br/>$\Rightarrow x=\frac{1}{2}$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\text { then } y^{2}=1-\left(\frac{1}{2}\right)^{2}=34 \\
<br/> <br/>\Rightarrow y=\frac{\sqrt{3}}{2}
<br/> <br/>\end{array}$
<br/> <br/>$\therefore$ The points of intersection are
<br/> <br/>$\mathrm{P}\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and $\mathrm{Q}\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$
<br/> <br/>It is clear from the figure that the shaded portion in region whose area is required.
<br/> <br/>$\therefore$ Required area = area OQAPO
<br/> <br/>$=2 \times$ area of the region OLAP
<br/> <br/>$=2 \times($ area of the region $O L P O+$ area of $L A P L$ )
<br/> <br/>$=2\left[\int_{0}^{\frac{1}{2}} \sqrt{1-(x-1)^{2}} d x+\int_{\frac{1}{2}}^{1} \sqrt{1-x^{2}} \mathrm{dx}\right]$
<br/> <br/>$=2\left[\frac{(x-1) \sqrt{1-(x-1)^{2}}}{2}+\frac{1}{2} \sin ^{-1}(x-1)\right]_{0}^{\frac{1}{2}}+2\left[\frac{x \sqrt{1-x^{2}}}{2}+\frac{1}{2} \sin ^{-1} x\right]_{\frac{1}{2}}^{1}$
<br/> <br/>$=-\frac{1}{2} \cdot \frac{\sqrt{3}}{2}+\sin ^{-1}\left(\frac{-1}{2}\right)-\sin ^{-1}(-1)+0+\sin ^{-1}(1)-\left(\frac{1}{2} \cdot \frac{\sqrt{3}}{2}+\sin ^{-1}\left(\frac{1}{2}\right)\right)$
<br/> <br/>$=\left(\frac{2 \pi}{3}-\frac{\sqrt{3}}{2}\right)$ sq. units.</div>


title: the-area-bounded-by-the-curve-y-sin-x-x-axis-and-the-ordinates-x-0-and-x-2-is
slug: the-area-bounded-by-the-curve-y-sin-x-x-axis-and-the-ordinates-x-0-and-x-2-is-85004
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The area bounded by the curve $y=\sin x, x$ -axis and the ordinates $x=0$ and $x=\pi / 2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\pi / 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Area $=\int_{0}^{\pi / 2} \mathrm{y} \mathrm{dx}=\int_{0}^{\pi / 2} \sin \mathrm{x} \mathrm{dx}=[-\cos \mathrm{x}]_{0}^{\pi / 2}=1$</div>


title: the-area-bounded-by-the-curves-x-2-y-2-0-and-x-3-y-2-1-is
slug: the-area-bounded-by-the-curves-x-2-y-2-0-and-x-3-y-2-1-is-89214
question: <div class="question">The area bounded by the curves $x+2 y^{2}=0$ and $x+3 y^{2}=1$ is</div>
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 sq. unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}$ sq. units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$ sq. units</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{4}{3}$ sq. units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{4}{3}$ sq. units</span> </div>

answer:
<div class="solution">We have, $x+2 y^{2}=0$
<br/> <br/>$\Rightarrow y^{2}=-\frac{x}{2} \ldots \text { (i) }$
<br/> <br/>a parabola with vertex $(0,0)$ and $x+3 y^{2}=1$
<br/> <br/>$\Rightarrow y^{2}=\frac{1-x}{3}=-\left(\frac{x-1}{3}\right) \ldots$ (ii), a parabola with vertex $(1,0)$
<br/> <br/>Solving (i) and (ii), we get $y=\pm 1$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/1uUb_Wysw315Rl0DFYCFG6Ly7ChrUi0sAteTJV3PPqo.original.fullsize.png"/><br/> <br/> <br/>$\mathrm{A}=\int_{-1}^{1}\left(\left(1-3 y^{2}\right)-\left(-2 y^{2}\right)\right) d y$
<br/> <br/>$=2 \int_{0}^{1}\left(1-y^{2}\right) d y=2\left[y-\frac{y^{3}}{3}\right]_{0}^{1}$
<br/> <br/>$=\frac{4}{3}$ sq.units</div>


tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
title:
the-area-bounded-by-the-x-axis-the-curve-y-f-x-and-the-lines-x-1-x-b-is-equal-to-b-2-1-2-for-all-b-1-then-f-x-is

slug:
the-area-bounded-by-the-x-axis-the-curve-y-f-x-and-the-lines-x-1-x-b-is-equal-to-b-2-1-2-for-all-b-1-then-f-x-is-78193

question:
<div class="question">The area bounded by the $x$ -axis, the curve $y=f(x)$ and the lines $\mathrm{x}=1, \mathrm{x}=\mathrm{b},$ is equal to $\sqrt{\mathrm{b}^{2}+1}-\sqrt{2}$
<br/> <br/>for all $b&gt;1,$ then $f(x)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{x-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{x+1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{x^{2}+1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{x}{\sqrt{1+x^{2}}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x}{\sqrt{1+x^{2}}}$</span> </div>

answer:
<div class="solution">Given $\int_{1}^{\mathrm{b}} \mathrm{f}(\mathrm{x}) \mathrm{dx}=\sqrt{\mathrm{b}^{2}+1}-\sqrt{2}$
<br/> <br/>Differentiate with respect to $\mathrm{b}$
<br/> <br/>$$
<br/> <br/>f(b)=\frac{b}{\sqrt{b^{2}+1}} \Rightarrow f(x)=\frac{x}{\sqrt{x^{2}+1}}
<br/> <br/>$$</div>


title: the-area-enclosed-between-the-curve-y-lo-g-e-x-e-and-the-coordinate-axes-is-1
slug: the-area-enclosed-between-the-curve-y-lo-g-e-x-e-and-the-coordinate-axes-is-1-82927
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The area enclosed between the curve $y=\log _e(x+e)$ and the coordinate axes is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ZM6sWNxcXPEqQvB8_byL2CiWCn7MvniGotvUdZThfPs.original.fullsize.png"/><br/><br/>Required area $A=\int_{1-e}^0 y d x=\int_{1-e}^0 \log _e(x+e) d x$<br/>put $x+e=t \Rightarrow d x=d t$ also when $x=1-e, t=$ 1 and when $x=0, t=e$<br/>$\therefore A=\int_1^e \log _e t d t=\left[t \log _e t-t\right]_1^e$<br/>$e-e-0+1=1$</div>


title: the-area-enclosed-by-the-curves-y-sin-x-cos-x-and-y-cos-x-sin-x-over-the-interval-0-2-is-1
slug: the-area-enclosed-by-the-curves-y-sin-x-cos-x-and-y-cos-x-sin-x-over-the-interval-0-2-is-1-32928
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0, \frac{\pi}{2}\right]$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$4(\sqrt{2}-1)$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$2 \sqrt{2}(\sqrt{2}-1)$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$2(\sqrt{2}+1)$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$2 \sqrt{2}(\sqrt{2}+1)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$2 \sqrt{2}(\sqrt{2}-1)$
<br/></span> </div>

answer:
<div class="solution">Given, $y=\sin x+\cos x, x \in[0, \pi / 2]$
<br/>$\frac{d y}{d x}=\cos x-\sin x$
<br/>$\begin{aligned} y &amp; =|\cos x-\sin x| \\ &amp; =\left[\begin{array}{cc}\cos x-\sin x &amp; x \in[0, \pi / 4] \\ \sin x-\cos x &amp; x \in[\pi / 4, \pi / 2]\end{array}\right]\end{aligned}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Bqoq95Rpm-j5ycIe8E3qOccP5zZI1leuKUG20Kyp-38.original.fullsize.png"/><br/> <br/>$\therefore$ Required area
<br/>$\begin{aligned}
<br/>&amp;=\int_0^{\pi / 4}|(\sin x+\cos x)-(\cos x-\sin x)| d x+\int_{\pi / 4}^{\pi / 2}|2 \cos x| d x
<br/>\end{aligned}$
<br/>$=\int_0^{\pi / 4}|2 \sin x| d x+\int_{\pi / 4}^{\pi / 2}|2 \cos x| d x$
<br/>$=2(-\cos x)_0^{\pi / 4}+2(\sin x)_{\pi / 4}^{\pi / 2}$
<br/>$=2\left[-\frac{1}{\sqrt{2}}+1+1-\frac{1}{\sqrt{2}}\right]$<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/h0i2UrjkiqoD78hh_1S802WjWVfCXjP7KBP5OpVWwrA.original.fullsize.png"/><br/> <br/>$\begin{aligned}= &amp; 2\left(2-\frac{2}{\sqrt{2}}\right) \\ = &amp; 2(2-\sqrt{2}) \\ = &amp; 4-2 \sqrt{2} \\ = &amp; 2 \sqrt{2}(\sqrt{2}-1)\end{aligned}$</div>


title: the-area-enclosed-by-the-curves-y-x-3-and-y-x-is
slug: the-area-enclosed-by-the-curves-y-x-3-and-y-x-is-37780
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area enclosed by the curves <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>3</mn></mfrac><mo> </mo><mi>sq</mi><mo> </mo><mi>units</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>4</mn></mfrac><mo> </mo><mi>sq</mi><mo> </mo><mi>units</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>12</mn></mfrac><mo> </mo><mi>sq</mi><mo> </mo><mi>units</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>12</mn><mn>5</mn></mfrac><mo> </mo><mi>sq</mi><mo> </mo><mi>units</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>12</mn></mfrac><mo> </mo><mi>sq</mi><mo> </mo><mi>units</mi></math></span> </div>

answer:
<div class="solution"><p>Since, the intersection of two curves</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/80f98405-f09a-4775-8881-61be77488333-image.png" style="width: 313px; height: 197px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>A</mi><mo>=</mo><mfenced close="|" open="|"><mrow><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><msqrt><mi>x</mi></msqrt></mrow></mfenced><mi>d</mi><mi>x</mi></mrow></mfenced><mo>=</mo><mfenced close="|" open="|"><msubsup><mfenced close="]" open="["><mrow><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>4</mn></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow><mn>3</mn></mfrac></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="|" open="|"><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced></mfenced><mo>=</mo><mfrac><mn>5</mn><mn>12</mn></mfrac><mo> </mo><mi>sq</mi><mo> </mo><mi>units</mi></math></p></div>


title: the-area-of-the-region-bounded-by-the-curve-yxx-xaxis-and-the-ordinates-x1-x1-is-given-by
slug: the-area-of-the-region-bounded-by-the-curve-yxx-xaxis-and-the-ordinates-x1-x1-is-given-by-45797
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area of the region bounded by the curve \(y=x|x|, x\)-axis and the ordinates \(x=1, x=-1\) is given by :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">zero</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{1}{3}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{2}{3}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{2}{3}\)</span> </div>

answer:
<div class="solution">The area of the region bounded by the curve \(y=f(x)\) and the ordinates \(x=a, x=b\) is given by<br/>\(\text {Area }=\left|\int_a^b y d x\right|\)<br/>According to the question,<br/>\(y=x|x|= \begin{cases}x^2, &amp; x \geq 0 \\ -x^2, &amp; x  &lt; 0\end{cases}\)<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/FZCw9yBYXriAVn3LRBhGUGiFnLoUl_9mYiLSqNxNlqc.original.fullsize.png"/><br/><br/>Required area<br/>\(=\) area of region \(\mathrm{OAB}+\) area of region \(\mathrm{OCD}\)<br/>\(=2 \times\) Area of region \(\mathrm{OAB}\)<br/>\(=2 \int_0^1 x^2 d x=\frac{2}{3} s q\). units</div>


tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P2.db
title:
the-area-of-the-region-bounded-by-the-parabola-y-2-2-x-1-the-tangent-to-the-parabola-at-the-point-2-3-and-the-x-axis-is-1

slug:
the-area-of-the-region-bounded-by-the-parabola-y-2-2-x-1-the-tangent-to-the-parabola-at-the-point-2-3-and-the-x-axis-is-1-40755

question:
<div class="question">The area of the region bounded by the parabola $(y-2)^2=(x-1)$, the tangent to the parabola at the point $(2,3)$ and the $X$-axis is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>3
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>6
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>9
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>12</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>9
<br/></span> </div>

answer:
<div class="solution">The equation of the parabola is $y^2-4 y-x+5=0$ The equation of tangent at $(2,3)$ is
<br/>$$
<br/>\begin{aligned}
<br/>3 y-2(y+3)-\frac{x+2}{2}+5 &amp; =0 \\
<br/>2 y-x-4 &amp; =0
<br/>\end{aligned}
<br/>$$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9b_OFOiPC-yESmoX1Lqx_VI1P1QvG9cP_LVngQLK6ac.original.fullsize.png"/><br/> <br/>$\therefore$ Required area $A$ is given by
<br/>$$
<br/>\begin{aligned}
<br/>&amp; A=\int_0^3\left(x_2-x_1\right) d y \\
<br/>&amp; \Rightarrow \quad A=\int_0^3\left[\left\{(y-2)^2+1\right\}-\{2 y-4\}\right] d y \\
<br/>&amp; \Rightarrow \quad A=\int_0^3\left(y^2-6 y+9\right) d y=\int_0^3(3-y)^2 d y \\
<br/>&amp; =-\left[\frac{(3-y)^3}{3}\right]_0^3=9 \\
<br/>&amp;
<br/>\end{aligned}
<br/>$$</div>


title: the-area-of-the-region-included-between-the-curves-x-2-y-2-a-2-and-x-y-a-a-0-is
slug: the-area-of-the-region-included-between-the-curves-x-2-y-2-a-2-and-x-y-a-a-0-is-63511
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area of the region included between the curves <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfenced close="|" open="|"><mi>x</mi></mfenced></msqrt><mo>+</mo><msqrt><mfenced close="|" open="|"><mi>y</mi></mfenced></msqrt><mo>=</mo><msqrt><mi>a</mi></msqrt><mfenced><mrow><mi>a</mi><mo>&gt;</mo><mn>0</mn></mrow></mfenced></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>π</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced><msup><mi>a</mi><mn>2</mn></msup></math> sq units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>-</mo><mi>π</mi></mrow></mfenced><msup><mi>a</mi><mn>2</mn></msup></math> sq units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup></math> sq units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>π</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced><msup><mi>a</mi><mn>2</mn></msup></math> sq units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>π</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced><msup><mi>a</mi><mn>2</mn></msup></math> sq units</span> </div>

answer:
<div class="solution"><p>Given curves are</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfenced close="|" open="|"><mi>x</mi></mfenced></msqrt><mo>+</mo><msqrt><mfenced close="|" open="|"><mi>y</mi></mfenced></msqrt><mo>=</mo><msqrt><mi>a</mi></msqrt><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/92c79c76-2fd3-47b1-8567-d7127f1e10c3-image.png" style="width: 234px; height: 270px;"/></p><p>Now, required areas <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math>[shaded area in the first quadrant]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfenced close="]" open="["><mrow><mfrac><mrow><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mi>a</mi></msubsup><msup><mfenced><mrow><msqrt><mi>a</mi></msqrt><mo>-</mo><msqrt><mi>x</mi></msqrt></mrow></mfenced><mn>2</mn></msup><mi>d</mi><mi>x</mi></mrow></mfenced></math>  [from Eqs. (i) and (ii)]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfenced close="]" open="["><mrow><mfrac><mrow><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mi>a</mi></msubsup><mfenced><mrow><mi>a</mi><mo>+</mo><mi>x</mi><mo>-</mo><mn>2</mn><msqrt><mi>a</mi></msqrt><msqrt><mi>x</mi></msqrt></mrow></mfenced><mi>d</mi><mi>x</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfrac><mrow><mi>π</mi><msup><mi>a</mi><mn>2</mn></msup></mrow><mn>4</mn></mfrac><mo>-</mo><mn>4</mn><msubsup><mfenced close="]" open="["><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><msqrt><mi>a</mi></msqrt><msup><mi>x</mi><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></mfenced><mn>0</mn><mi>a</mi></msubsup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>π</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced><msup><mi>a</mi><mn>2</mn></msup></math> sq units</p><p>Hence, option (a) is correct.</p></div>


title: the-area-of-the-region-r-x-y-5-x-2-y-2-x-2-9-is-1
slug: the-area-of-the-region-r-x-y-5-x-2-y-2-x-2-9-is-1-98926
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area of the region<br/>$\mathbf{R}=\left\{(x, y): 5 x^2 \leq y \leq 2 x^2+9\right\} \text { is : }$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$11 \sqrt{3}$ square units</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$12 \sqrt{3}$ square units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9 \sqrt{3}$ square units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$6 \sqrt{3}$ square units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$12 \sqrt{3}$ square units</span> </div>

answer:
<div class="solution"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/TYcOVCHhyD7Ep5QjIfRNd4Z-BVnCVVqI_MsguvXZzqQ.original.fullsize.png"/><br/><br/>$\begin{aligned} &amp; \text { Required area }=2 \int_0^{\sqrt{3}}\left(2 x^2+9-5 x^2\right) d x \\ &amp; =2\left[9 x-x^3\right]_0^{\sqrt{3}}=2[9 \sqrt{3}-3 \sqrt{3}]=12 \sqrt{3} .\end{aligned}$</div>


title: the-area-of-the-region-r-x-y-x-y-and-x-2-y-2-1-is-1
slug: the-area-of-the-region-r-x-y-x-y-and-x-2-y-2-1-is-1-47339
tags: ['Mathematics', 'Area Under Curves', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area of the region $\mathrm{R}=\left\{(x, y):|x| \leq|y|\right.$ and $x^{2}$ $\left.+y^{2} \leq 1\right\}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3 \pi}{8}$ sq units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5 \pi}{8} \mathrm{sq}$ units</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{2}$ sq units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{8}$ sq unit</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{2}$ sq units</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5nCmf5KCGMlyNK8_pS9tpwndw7e6OZXR_vf-JKknoYU.original.fullsize.png"/><br/> <br/> <br/>Required area $=4$ (Area of the shaded region in first quadrant)
<br/> <br/>$$
<br/> <br/>=4 \int_{0}^{1 \sqrt{2}}\left(y_{1}-y_{2}\right) d x=4 \int_{0}^{1 \sqrt{2}}\left(\sqrt{1-x^{2}}-x\right) d x
<br/> <br/>$$
<br/> <br/>$=4\left[\frac{x}{2} \sqrt{1-x^{2}}+\frac{1}{2} \sin ^{-1} x-\frac{x^{2}}{2}\right]_{0}^{1 / \sqrt{2}}$
<br/> <br/>$=4\left[\frac{1}{2 \sqrt{2}} \times \frac{1}{\sqrt{2}}+\frac{1}{2} \times \frac{\pi}{4}-\frac{1}{4}\right]$
<br/> <br/>$=4\left[\frac{1}{4}+\frac{\pi}{8}-\frac{1}{4}\right]=\frac{4 \pi}{8}=\frac{\pi}{2}$ sq units</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-area-of-the-triangle-formed-by-joining-the-origin-to-the-point-of-intersection-of-the-line-x-5-2-y-3-5-and-circle-x-2-y-2-10-is

slug:
the-area-of-the-triangle-formed-by-joining-the-origin-to-the-point-of-intersection-of-the-line-x-5-2-y-3-5-and-circle-x-2-y-2-10-is-62506

question:
<div class="question">The area of the triangle formed by joining the origin to the point of intersection of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msqrt><mn>5</mn></msqrt><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>3</mn><msqrt><mn>5</mn></msqrt></math> and circle <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>10</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> </li> </ul>

answer:
<div class="solution"><p>Length of perpendicular from origin to the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msqrt><mn>5</mn></msqrt><mo>+</mo><mn>2</mn><mi>y</mi><mo>=</mo><mn>3</mn><msqrt><mn>5</mn></msqrt></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>L</mi><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow><msqrt><msup><mfenced><msqrt><mn>5</mn></msqrt></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup></msqrt></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>5</mn></msqrt></mrow><msqrt><mn>9</mn></msqrt></mfrac><mo>=</mo><msqrt><mn>5</mn></msqrt></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6fa9b97b-9eb2-4746-8765-3fb218060d92-d46.png" style="width: 294px; height: 230px;"/></p><p>Radius of the given circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>10</mn></msqrt><mo>=</mo><mi>O</mi><mi>Q</mi><mo>=</mo><mi>O</mi><mi>P</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>=</mo><mn>2</mn><mi>Q</mi><mi>L</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msqrt><mi>O</mi><msup><mi>Q</mi><mn>2</mn></msup><mo>-</mo><mi>O</mi><msup><mi>L</mi><mn>2</mn></msup></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msqrt><mn>10</mn><mo>-</mo><mn>5</mn></msqrt><mo>=</mo><mn>2</mn><msqrt><mn>5</mn></msqrt></math></p><p>Thus, area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>O</mi><mi>P</mi><mi>Q</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mi>P</mi><mi>Q</mi><mo>×</mo><mi>O</mi><mi>L</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>2</mn><msqrt><mn>5</mn></msqrt><mo>×</mo><msqrt><mn>5</mn></msqrt><mo>=</mo><mn>5</mn></math> sq units</p><p>Hence, option (c) is correct.</p></div>


title: the-area-under-the-curve-y-cos-x-sin-x-0-x-2-and-above-x-axis-is-1
slug: the-area-under-the-curve-y-cos-x-sin-x-0-x-2-and-above-x-axis-is-1-91301
tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The area under the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced close="|" open="|"><mrow><mi>cos</mi><mi>x</mi><mo>-</mo><mi>sin</mi><mi>x</mi></mrow></mfenced><mo>,</mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math> and above <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-axis, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Required area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mfenced close="|" open="|"><mrow><mi>cos</mi><mi>x</mi><mo>-</mo><mi>sin</mi><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>π</mi><mo>/</mo><mn>4</mn></mrow></msubsup><mfenced><mrow><mi>cos</mi><mi>x</mi><mo>-</mo><mi>sin</mi><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>π</mi><mo>/</mo><mn>4</mn></mrow><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow></msubsup><mfenced><mrow><mi>sin</mi><mi>x</mi><mo>-</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced close="]" open="["><mrow><mi>sin</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mi>x</mi></mrow></mfenced><mn>0</mn><mrow><mi>π</mi><mo>/</mo><mn>4</mn></mrow></msubsup><mo>+</mo><msubsup><mfenced close="]" open="["><mrow><mo>-</mo><mi>cos</mi><mi>x</mi><mo>-</mo><mi>sin</mi><mi>x</mi></mrow></mfenced><mrow><mi>π</mi><mo>/</mo><mn>4</mn></mrow><mrow><mi>π</mi><mo>/</mo><mn>2</mn></mrow></msubsup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfenced><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn></math></p></div>


tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-arithmetic-mean-of-a-set-of-observation-is-x-if-each-observation-is-divided-by-and-increased-by-10-then-the-mean-of-the-new-series-is

slug:
the-arithmetic-mean-of-a-set-of-observation-is-x-if-each-observation-is-divided-by-and-increased-by-10-then-the-mean-of-the-new-series-is-12390

question:
<div class="question">The arithmetic mean of a set of observation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>¯</mo></mover></math>. If each observation is divided by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and increased by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>, then the mean of the new series is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mover><mi>X</mi><mo>¯</mo></mover><mi>α</mi></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mi>X</mi><mo>¯</mo></mover><mo>+</mo><mn>10</mn></mrow><mi>α</mi></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mi>X</mi><mo>¯</mo></mover><mo>+</mo><mn>10</mn><mi>α</mi></mrow><mi>α</mi></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mover><mi>X</mi><mo>¯</mo></mover><mo>+</mo><mn>10</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mover><mi>X</mi><mo>¯</mo></mover><mo>+</mo><mn>10</mn><mi>α</mi></mrow><mi>α</mi></mfrac></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> observations.</p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mi>Σ</mi><msub><mi>x</mi><mi>i</mi></msub></math></p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mfrac><msub><mi>x</mi><mi>i</mi></msub><mi>α</mi></mfrac><mo>+</mo><mn>10</mn></math></p><p>Then. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>α</mi></mfrac><mfenced><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac><mi>Σ</mi><msub><mi>x</mi><mi>i</mi></msub></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mfenced><mrow><mn>10</mn><mi>n</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mover><mi>X</mi><mo>¯</mo></mover><mtext>new</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mi>α</mi></mfrac><mover><mi>X</mi><mo>¯</mo></mover><mo>+</mo><mn>10</mn><mo>=</mo><mfrac><mrow><mover><mi>X</mi><mo>¯</mo></mover><mo>+</mo><mn>10</mn><mi>α</mi></mrow><mi>α</mi></mfrac></math></p><p>Hence option (c) is correct.</p></div>


title: the-arithmetic-mean-of-numbers-a-b-c-d-e-is-m-what-is-the-value-of-a-m-b-m-c-m-d-m-e-m-1
slug: the-arithmetic-mean-of-numbers-a-b-c-d-e-is-m-what-is-the-value-of-a-m-b-m-c-m-d-m-e-m-1-96102
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The arithmetic mean of numbers a, $\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}$ is $\mathrm{M}$. What is the value of $(a-M)+(b-M)+(c-M)+$ $(\mathrm{d}-\mathrm{M})+(\mathrm{e}-\mathrm{M}) ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{M}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a+b+c+d+e$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 \mathrm{M}$</span> </li> </ul>

answer:
<div class="solution">Given $\mathrm{M}=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}}{5}$
<br/> <br/>$\Rightarrow a+b+c+d+e=5 M$
<br/> <br/>$\Rightarrow a+b+c+d+e-5 m=0$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow(a-M)+(b-M)+(c-M)+(d-M) \\
<br/> <br/>+(e-M)=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Hence, required value $=0$</div>


title: the-arithmetic-mean-of-the-data-0-1-2-n-with-frequencies-1-4-c-1-4-c-2-c-a-is
slug: the-arithmetic-mean-of-the-data-0-1-2-n-with-frequencies-1-4-c-1-4-c-2-c-a-is-13802
tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
question:
<div class="question">The arithmetic mean of the data $0,1,2, \ldots \ldots, n$ with frequencies $1,{ }^{4} \mathrm{C}_{1}{ }^{4} \mathrm{C}_{2}, \ldots . .,{ }^{\infty} \mathrm{C}_{\mathrm{a}}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{n}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2^{n}}{n}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{n}+1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{n}}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{n}}{2}$</span> </div>

answer:
<div class="solution">Since, Mean $=\frac{\sum \mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}}{\sum \mathrm{f}_{\mathrm{i}}}$ where $\mathrm{x}_{\mathrm{i}}$ are
<br/> <br/>observations with frequencies $\mathrm{f}_{\mathrm{i}}, \mathrm{i}=1,2,$ $\ldots \ldots \ldots$
<br/> <br/>The required mean is given by
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\overline{\mathrm{X}} &amp;=\frac{0.1+1 .{ }^{\mathrm{n}} \mathrm{C}_{1}+2{ }^{\mathrm{n}} \mathrm{C}_{2}+\ldots \ldots+\mathrm{n} .{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}}{1+{ }^{\mathrm{n}} \mathrm{C}_{1}+{ }^{\mathrm{n}} \mathrm{C}_{2}+\ldots+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}}} \\
<br/> <br/>=&amp; \frac{\sum_{\mathrm{r}=0}^{\mathrm{n}} \mathrm{r}^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}{\sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}=\frac{\sum_{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r} \cdot \frac{\mathrm{n}}{\mathrm{r}}{ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}}{\sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}} \\
<br/> <br/>=&amp; \frac{\mathrm{n} \sum{\sum}^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}}{\sum_{\mathrm{r}=0}^{\mathrm{n}}{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}=\frac{\mathrm{n} .2^{\mathrm{n}-1}}{2^{\mathrm{n}}}=\frac{\mathrm{n}}{2}
<br/> <br/>\end{aligned}
<br/> <br/>$$</div>


title: the-atomic-radius-of-atom-is-r-total-volume-of-atoms-present-in-a-f-cc-unit-cell-of-an-element-is
tags: ['Chemistry', 'Solid State', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
slug:
the-atomic-radius-of-atom-is-r-total-volume-of-atoms-present-in-a-f-cc-unit-cell-of-an-element-is-96206

question:
<div class="question">The atomic radius of atom is r. Total volume of atoms present in a $f c c$ unit cell of an element is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{24}{3} \pi r^{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{12}{3} \pi r^{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{16}{3} \pi r^{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{16}{3} \pi r^{3}$</span> </div>

answer:
<div class="solution">4 atom are present in $f c c$.<br/>$$<br/>\mathrm{So}, \mathrm{V}=4\left[\frac{4}{3} \pi \mathrm{r}^{3}\right]=\frac{16}{3} \pi \mathrm{r}^{3}<br/>$$</div>


title: the-average-kinetic-energy-of-an-ideal-gas-per-molecule-in-si-unit-at-2-5-c-will-be
slug: the-average-kinetic-energy-of-an-ideal-gas-per-molecule-in-si-unit-at-2-5-c-will-be-93744
tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The average kinetic energy of an ideal gas per molecule in SI unit at $25^{\circ} \mathrm{C}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6.17 \times 10^{-21} \mathrm{~kJ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$6.17 \times 10^{-21} \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$6.17 \times 10^{-20} \mathrm{~J}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7.16 \times 10^{-20} \mathrm{~J}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6.17 \times 10^{-21} \mathrm{~J}$</span> </div>

answer:
<div class="solution">$\overline{\mathrm{KE}}=\frac{3}{2} \mathrm{kT}=\frac{3}{2} \times \frac{8.313}{6.023 \times 10^{23}} \times 298=6.17 \times 10^{-21} \mathrm{~J}$
<br/> <br/>(Average Kinetic energy $\left.\overline{\mathrm{KE}}=\frac{3}{2} \mathrm{k} \mathrm{T}=\frac{3}{2} \frac{\mathrm{R}}{\mathrm{N}} \mathrm{T}\right)$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.048</span> </div>
source_db: MarksBatch2_P2.db
title:
the-average-translational-kinetic-energy-of-o-2-molar-mass-32-molecules-at-a-particular-temperature-is-0048-ev-the-translational-kinetic-energy-of-n-2

slug:
the-average-translational-kinetic-energy-of-o-2-molar-mass-32-molecules-at-a-particular-temperature-is-0048-ev-the-translational-kinetic-energy-of-n-2-82144

question:
<div class="question">The average translational kinetic energy of $\mathrm{O}_{2}$ (molar mass 32) molecules at a particular temperature is $0.048 \mathrm{eV}$. The translational kinetic energy of $\mathrm{N}_{2}$ (molar mass 28 ) molecules in $\mathrm{eV}$ at the same temperature is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.0015</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.003</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0.048</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.768</span> </li> </ul>

answer:
<div class="solution">Given,
<br/>\(\mathrm{E}_{\mathrm{O}_{2}}=0.048 \mathrm{eV}\)
<br/>We know that,
<br/>Translational kinetic energy,
<br/>\(\mathrm{E}=\frac{3}{2} \mathrm{KT}\)
<br/>where, \(\mathrm{k}=\) Boltzmann constant
<br/>\(\mathrm{T}=\) Temperature
<br/>\(\therefore \mathrm{E} \propto \mathrm{T}\)
<br/>Since, the temperature is same for both oxygen and nitrogen
<br/>therefore, \(\mathrm{E}_{\mathrm{O}_{2}}=\mathrm{E}_{\mathrm{N}_{2}}\)
<br/>\(\therefore \mathrm{E}_{\mathrm{N}_{2}}=0.048 \mathrm{eV}\)</div>


tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
title:
the-back-emf-induced-in-a-coil-when-current-changes-from-1-ampere-to-zero-in-one-millisecond-is-4-volts-the-self-inductance-of-the-coil-is

slug:
the-back-emf-induced-in-a-coil-when-current-changes-from-1-ampere-to-zero-in-one-millisecond-is-4-volts-the-self-inductance-of-the-coil-is-98199

question:
<div class="question">The back emf induced in a coil, when current changes from 1 ampere to zero in one milli-second, is 4 volts, the self inductance of the coil is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 henry</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4 henry</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10^{-3}$ henry</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 \times 10^{-3}$ henry</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \times 10^{-3}$ henry</span> </div>

answer:
<div class="solution">$\mathrm{e}=-\mathrm{L} \frac{\mathrm{di}}{\mathrm{dt}}$<br/>But $\mathrm{e}=4 \mathrm{~V}$ and $\frac{\mathrm{di}}{\mathrm{dt}}=\frac{0-1}{10^{-3}}=-1 / 10^{-3}$<br/>$\therefore \frac{-1}{10^{-3}}(-\mathrm{L})=4 \Rightarrow \mathrm{L}=4 \times 10^{-3}$ henry</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
title:
the-beam-of-light-has-three-wavelengths-4144-a-4972-a-and-6216-a-with-a-total-intensity-of-36-1-0-3-wm-2-equally-distributed-amongst-the-three-wavelen

slug:
the-beam-of-light-has-three-wavelengths-4144-a-4972-a-and-6216-a-with-a-total-intensity-of-36-1-0-3-wm-2-equally-distributed-amongst-the-three-wavelen-14025

question:
<div class="question">The beam of light has three wavelengths $4144 Ã, 4972 Ã$ and $6216 Ã$ with a total intensity of $3.6 \times 10^{-3} \mathrm{Wm}^{2}$ equally distributed amongst the three wavelengths. The beam falls normally on the area $1 \mathrm{~cm}^{2}$ of a clean metallic surface of work function $2.3 \mathrm{eV}$. Assume that there is $\mathrm{n} 0$ loss of light by reflection and that each energetically capable photon ejects one electron. Calculate the number of photoelectrons liberated in $2 \mathrm{~s}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \times 10^{9}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.075 \times 10^{12}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9 \times 10^{8}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3.75 \times 10^{6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.075 \times 10^{12}$</span> </div>

answer:
<div class="solution">As we know, threshold wavelength
<br/> <br/>$$
<br/> <br/>\begin{array}{c}
<br/> <br/>\left(\lambda_{0}\right)=\frac{\mathrm{hc}}{\phi} \\
<br/> <br/>\Rightarrow \lambda_{0}=\frac{\left(6.63 \times 10^{-34}\right) \times 3 \times 10^{8}}{2.3 \times\left(1.6 \times 10^{-19}\right)}=5.404 \times 10^{-7} \mathrm{~m}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$\Rightarrow \lambda_{0}=5404 Ãâ¦$
<br/> <br/>Hence, wavelength $4144 Ãâ¦$ and $4972 Ãâ¦$ will emit electron from the metal surface. For each wavelength energy incident on the surface per unit time = intensity of each $\times$ area of the surface wavelength
<br/> <br/>$$
<br/> <br/>=\frac{3.6 \times 10^{-3}}{3} \times(\mathrm{lcm})^{2}=1.2 \times 10^{-7} \text {joule }
<br/> <br/>$$
<br/> <br/>Therefore, energy incident on the surface for each wavelength in $2 \mathrm{~s}$
<br/> <br/>$$
<br/> <br/>\mathrm{E}=\left(1.2 \times 10^{-7}\right) \times 2=2.4 \times 10^{-7} \mathrm{~J}
<br/> <br/>$$
<br/> <br/>Number of photons $\mathrm{n}_{1}$ due to wavelength $4144 Ãâ¦$
<br/> <br/>$$
<br/> <br/>n_{1}=\frac{\left(2.4 \times 10^{-7}\right)\left(4144 \times 10^{-10}\right)}{\left(6.63 \times 10^{-34}\right)\left(3 \times 10^{8}\right)}=0.5 \times 10^{12}
<br/> <br/>$$
<br/> <br/>Number of photons $\mathrm{n}_{2}$ due to the wavelength $4972 Ãâ¦$
<br/> <br/>$$
<br/> <br/>\mathrm{n}_{2}=\frac{\left(2.4 \times 10^{-7}\right)\left(4972 \times 10^{-10}\right)}{\left(6.63 \times 10^{-34}\right)\left(3 \times 10^{8}\right)}=0.572 \times 10^{12}
<br/> <br/>$$
<br/> <br/>Therefore total number of photoelectrons liberated in $2 \mathrm{~s}$
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\mathrm{N} &amp;=\mathrm{n}_{1}+\mathrm{n}_{2} \\
<br/> <br/>&amp;=0.5 \times 10^{12}+0.575 \times 10^{12} \\
<br/> <br/>&amp;=1.075 \times 10^{12}
<br/> <br/>\end{aligned}
<br/> <br/>$$</div>


tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-bob-of-simple-pendulum-is-a-spherical-hollow-ball-filled-with-water-a-plugged-hole-near-the-bottom-of-the-oscillating-bob-get-suddenly-unplugged-d

slug:
the-bob-of-simple-pendulum-is-a-spherical-hollow-ball-filled-with-water-a-plugged-hole-near-the-bottom-of-the-oscillating-bob-get-suddenly-unplugged-d-45150

question:
<div class="question">The bob of simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob get suddenly unplugged. During observation, till water is coming out, the time period of oscillation would</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">first increase and then decrease to the original value</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">first decrease and then increase to the original value</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">remain unchanged</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">increase towards a saturation value</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">first increase and then decrease to the original value</span> </div>

answer:
<div class="solution"><p>The bob filled completely with water has its centre of mass at its centre. The time period of oscillation is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>2</mn><mi>π</mi><msqrt><mfrac><mi>l</mi><mi>g</mi></mfrac></msqrt></math></p><p>As the water starts coming out of the bob, its centre of mass shifts vertically downward and as a result, effective length of the pendulum increases and hence its period also increases. When the bob is empty, again its centre of mass appears at its centre and as a result, the period of oscillation again reaches to its original value.</p></div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2.12</span> </div>
source_db: MarksBatch2_P2.db
title:
the-bohr-orbit-radius-for-the-hydrogen-atom-n-1-is-approximately-0530-a-the-radius-for-the-first-excited-state-n-2-orbit-is-in-a-1

slug:
the-bohr-orbit-radius-for-the-hydrogen-atom-n-1-is-approximately-0530-a-the-radius-for-the-first-excited-state-n-2-orbit-is-in-a-1-99956

question:
<div class="question">The Bohr orbit radius for the hydrogen atom $(n=1)$ is approximately $0.530 Ã$. The radius for the first excited state $(n=2)$ orbit is (in $Ã$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.13</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.06</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4.77</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">2.12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given : Radius of hydrogen atom $=0.530$<br/>$Ã$, Number of excited state $(n)=2$ and atomic number of hydrogen atom $(Z)=1$. We know that the Bohr radius<br/>$(r)=\frac{n^2}{Z} \times \text { radius of atom }=\frac{(2)^2}{1} \times 0.530$<br/>$=4 \times 0.530=2.12 Ã$</div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2.12</span> </div>
source_db: MarksBatch2_P2.db
title:
the-bohr-orbit-radius-for-the-hydrogen-atom-n-1-is-approximately-0530-a-the-radius-for-the-first-excited-state-n-2-orbit-is-in-a

slug:
the-bohr-orbit-radius-for-the-hydrogen-atom-n-1-is-approximately-0530-a-the-radius-for-the-first-excited-state-n-2-orbit-is-in-a-53310

question:
<div class="question">The Bohr orbit radius for the hydrogen atom $(n=1)$ is approximately $0.530 Ã$. The radius for the first excited state $(n=2)$ orbit is $(\operatorname{in} Ã)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.13</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1.06</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4.77</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">2.12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Radius of hydrogen atom $=0.530 Ãâ¦$, Number of excited state $(n)=2$ and atomic number of hydrogen atom $(Z)=1$. We know that the Bohr radius.
<br/> <br/>$$
<br/> <br/>(r)=\frac{n^{2}}{Z} \times \text { Radius of atom }=\frac{(2)^{2}}{1} \times 0.530
<br/> <br/>$$
<br/> <br/>$=4 \times 0.530=2.12 Ãâ¦$</div>


title: the-boiling-point-of-alkyl-halide-are-higher-than-those-of-corresponding-alkanes-because-of
slug: the-boiling-point-of-alkyl-halide-are-higher-than-those-of-corresponding-alkanes-because-of-97316
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
question:
<div class="question">The boiling point of alkyl halide are higher than those of corresponding alkanes because of</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">dipole-dipole interaction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">dipole-induced dipole interaction</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">H-bonding</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">dipole-dipole interaction</span> </div>

answer:
<div class="solution">Due to dipole-dipole interaction the boiling point of alkyl halide is higher as compared to corresponding alkanes.</div>


title: the-boiling-point-of-water-is-exceptionally-high-because
slug: the-boiling-point-of-water-is-exceptionally-high-because-30118
question: <div class="question">The boiling point of water is exceptionally high because</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2012']
answer: <div class="solution">The high boiling point of water is due to H-bonding.</div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">there is a covalent bond between $\mathrm{H}$ and $\mathrm{O}$.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">water molecule is linear.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">water molecules associate due to hydrogen bonding.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">water molecule is not linear.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">water molecules associate due to hydrogen bonding.</span> </div>


title: the-by-product-of-solvayammonia-process-is
slug: the-by-product-of-solvayammonia-process-is-41331
question: <div class="question">The by product of Solvay-ammonia process is</div>
tags: ['Chemistry', 's Block Elements', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CO}_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{NH}_{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{CaCl}_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CaCO}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CaCl}_{2}$</span> </div>

answer:
<div class="solution">$\mathrm{CaCl}_{2}$ is produced as a byproduct in solvay ammonia process.
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\text { (i) } \mathrm{NaCl}+\mathrm{CO}_{2}+\mathrm{NH}_{3}+\mathrm{H}_{2} \mathrm{O} \\
<br/> <br/>\longrightarrow \mathrm{NaHCO}_{3}+\mathrm{NH}_{4} \mathrm{Cl}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>(I) $\mathrm{CaCO}_{3} \longrightarrow \mathrm{CO}_{2}+\mathrm{CaO}$
<br/> <br/> $2 \mathrm{NH}_{2} \mathrm{Cl}+\mathrm{CaO} \longrightarrow 2 \mathrm{NH}_{3}+$
<br/> <br/>$$
<br/> <br/>\mathrm{CaCl}_{2}+\mathrm{H}_{2} \mathrm{C}
<br/> <br/>$$
<br/> <br/>By podust</div>


title: the-carbylamine-reaction-is-1
slug: the-carbylamine-reaction-is-1-30075
question: <div class="question">The carbylamine reaction is</div>
tags: ['Chemistry', 'Amines', 'JEE Main']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/EVuJWcyyAryG9k6z1moOv-NF-ugTwyHzXu10IlCNStk.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Y1xRv6XaOvMfxarWorG3VgIYP2D00f8_2AjBHD2tuis.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/7K2B8UfzAjEimME7HOo9doc9YelfoU9QVDxM0Cs-7Tk.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/4ZDV-C1_YPnp1w-ET_JO-ofupUOA4a-JhJK_fDDpe50.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Y1xRv6XaOvMfxarWorG3VgIYP2D00f8_2AjBHD2tuis.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Primary amines (aromatic or aliphatic) on warming with chloroform and alcoholic KOH, gives carbylamine having offensive smell. This reaction is called carbylamine reaction.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/XOcOrHmBhgzAeKgYlWvaoHflmlTNtB0X0uz61zdqJ4U.original.fullsize.png"/></div>


title: the-chemical-reaction-2-o-3-3-o-2-proceeds-as-follows-rate-law-expression-should-be-1
slug: the-chemical-reaction-2-o-3-3-o-2-proceeds-as-follows-rate-law-expression-should-be-1-85047
tags: ['Chemistry', 'Chemical Kinetics', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">The chemical reaction $2 \mathrm{O}_{3} \longrightarrow 3 \mathrm{O}_{2}$ proceeds as follows:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/q-k5qeKWP9rRCW7R4z4KEsX-sdRQd33HtNSxGsuFyD4.original.fullsize.png"/><br/><br/>rate law expression should be:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{r}=\mathrm{k}\left[\mathrm{O}_{3}\right]^{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{r}=\mathrm{k}\left[\mathrm{O}_{3}\right]^{2}\left[\mathrm{O}_{2}\right]^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{r}=\mathrm{k}\left[\mathrm{O}_{3}\right]^{2}\left[\mathrm{O}_{2}\right]^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{r}=\mathrm{k}\left[\mathrm{O}_{3}\right]\left[\mathrm{O}_{2}\right]^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{r}=\mathrm{k}\left[\mathrm{O}_{3}\right]^{2}\left[\mathrm{O}_{2}\right]^{-1}$</span> </div>

answer:
<div class="solution">$\mathrm{O}_{3} \stackrel{\text { Fast }}{\longrightarrow} \mathrm{O}_{2}+\mathrm{O}$<br/>$$<br/>\begin{array}{l}<br/>\mathrm{O}+\mathrm{O}_{3} \stackrel{\text { Slow }}{\longrightarrow} 2 \mathrm{O}_{2} \\<br/>\mathrm{k}=\frac{\left[\mathrm{O}_{2}\right][\mathrm{O}]}{\left[\mathrm{O}_{3}\right]} \\<br/>\text { Rate }=\mathrm{k}^{\prime}\left[\mathrm{O}_{3}\right][\mathrm{O}]<br/>\end{array}<br/>$$<br/>put [O] from (ii)<br/>$$<br/>\mathrm{r}=\frac{\mathrm{k}^{\prime}\left[\mathrm{O}_{3}\right] \mathrm{k}\left[\mathrm{O}_{3}\right]}{\left[\mathrm{O}_{2}\right]}=\mathrm{k}\left[\mathrm{O}_{3}\right]^{2}\left[\mathrm{O}_{2}\right]^{-1}<br/>$$<br/>Note: Intermediates are never represented in rate law equation.</div>


title: the-circuit-is-equivalent-to-2
slug: the-circuit-is-equivalent-to-2-17332
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The circuit is equivalent to
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/vqymBaCvABIHeItSBnvLirbFMdZc6HFR5hwbqpZuYgE.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">AND gate</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">OR gate</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Not gate</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">OR gate</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{Y} &amp;=\mathrm{A}+(\overline{\mathrm{A}} \cdot \mathrm{B})=(\mathrm{A}+\overline{\mathrm{A}}) \cdot(\mathrm{A}+\mathrm{B}) \\ &amp;=\mathrm{A}+\mathrm{B} \\ \Rightarrow &amp; \mathrm{OR} \text { gate } \end{aligned}$</div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
title:
the-coefficient-of-friction-between-the-rubber-tyres-and-the-road-wayis-025-the-maximum-speed-with-which-a-car-can-be-driven-round-a-curve-of-radius-2

slug:
the-coefficient-of-friction-between-the-rubber-tyres-and-the-road-wayis-025-the-maximum-speed-with-which-a-car-can-be-driven-round-a-curve-of-radius-2-91588

question:
<div class="question">The coefficient of friction between the rubber tyres and the road wayis $0.25 .$ The maximum speed with which a car can be driven round a curve of radius $20 \mathrm{~m}$ without skidding is $\left(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$7 \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$14 \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7 \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">$\mu \mathrm{mg}=\mathrm{mv}^{2} / \mathrm{r}$ or $\mathrm{v}=\sqrt{\mu \mathrm{g} \mathrm{r}}$
<br/> <br/>or $v=\sqrt{(0.25 \times 9.8 \times 20)}=7 \mathrm{~m} / \mathrm{s}$</div>


title: the-coefficient-of-the-middle-term-in-the-expansion-of-2-3-x-4-is
slug: the-coefficient-of-the-middle-term-in-the-expansion-of-2-3-x-4-is-21584
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">216</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of the middle term in the expansion of $(2+3 \mathrm{x})^{4}$ is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 !$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 !$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">216</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">When exponent is $n$ then total number of terms are $\mathrm{n}+1 .$ So, total number of terms in $(2+3 x)^{4}=5$
<br/> <br/>Middle term is 3rd. $\Rightarrow \mathrm{T}_{3}={ }^{4} \mathrm{C}_{2}(2)^{2} \cdot(3 \mathrm{x})^{2}$
<br/> <br/>$$
<br/> <br/>=\frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2} \times 4 \times 9 \mathrm{x}^{2}=216 \mathrm{x}^{2}
<br/> <br/>$$</div>


title: the-coefficient-of-the-term-independent-of-x-in-the-expansion-of-3-x-2-x-2-3-10-is
slug: the-coefficient-of-the-term-independent-of-x-in-the-expansion-of-3-x-2-x-2-3-10-is-38651
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of the term independent of $x$ in the expansion of $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{5}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{9}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{4}$</span> </div>

answer:
<div class="solution">The $(r+1)$ th term in the expansion of $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}$ is given by
<br/> <br/>$\begin{array}{l}
<br/> <br/>\mathrm{T}_{r+1}={ }^{10} \mathrm{C}_{r}\left(\sqrt{\frac{x}{3}}\right)^{10-r}\left(\frac{3}{2 x^{2}}\right)^{r} \\
<br/> <br/>={ }^{10} \mathrm{C}_{r} \frac{x^{5-\left(\frac{r}{2}\right)}}{3^{5-\left(\frac{r}{2}\right)}} \cdot \frac{3^{r}}{2^{r} x^{2 r}} \\
<br/> <br/>={ }^{10} \mathrm{C}_{r} \frac{3^{\left(\frac{3 r}{2}\right)-5}-5-\left(\frac{5 r}{2}\right)}{2^{r}} x
<br/> <br/>\end{array}$
<br/> <br/>$\mathrm{T}_{r+1}={ }^{10} \mathrm{C}_{r}\left(\sqrt{\frac{x}{3}}\right)^{10-r}\left(\frac{3}{2 x^{2}}\right)^{r}$ $={ }^{10} \mathrm{C}_{r} \frac{x^{5-\left(\frac{r}{2}\right)}}{3} \cdot \frac{3^{r}}{2^{r} x^{2 r}}$ $={ }^{10} \mathrm{C}_{r} \frac{3^{\left(\frac{3 r}{2}\right)-5}-\left(\frac{5}{2}\right)-\left(\frac{5 r}{2}\right)}{2^{r}} x$ For $\mathrm{T}_{r+1}$ to be independent of $x$, we must have $5-\left(\frac{5 r}{2}\right)=0$ or $r=2$.
<br/> <br/>Thus, the 3 rd term is independent of $x$ and is equal to ${ }^{10} \mathrm{C}_{2} \frac{3^{3-5}}{2^{2}}=\frac{10 \times 9}{2} \times \frac{3^{-2}}{4}=\frac{5}{4}$</div>


title: the-coefficient-of-x-4-in-the-expansion-of-1-x-x-2-x-3-11-is
slug: the-coefficient-of-x-4-in-the-expansion-of-1-x-x-2-x-3-11-is-21811
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">990</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of $\mathrm{x}^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{11},$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">440</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">770</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">990</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1001</span> </li> </ul>

answer:
<div class="solution">We have coefficient of $x^{4}$ in $\left(1+x+x^{2}+x^{3}\right)^{11}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=\text { coefficient of } \mathrm{x}^{4} \text { in }\left(1+\mathrm{x}^{2}\right)^{11}(1+\mathrm{x})^{11} \\
<br/> <br/>=\text { coefficient of } \mathrm{x}^{4} \text { in }(1+\mathrm{x})^{11}+\text { coefficient } \\
<br/> <br/>\text { of } \mathrm{x}^{2} \text { in } \\
<br/> <br/>\text { 11. }(1+\mathrm{x})^{11}+\text { constant term is }{ }^{11} \mathrm{C}_{2} \cdot(1+\mathrm{x})^{11} \\
<br/> <br/>={ }^{11} \mathrm{C}_{4}+11 .{ }^{11} \mathrm{C}_{2}+{ }^{11} \mathrm{C}_{2}=990
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: the-coefficient-of-x-5-in-the-expansion-of-1-x-21-1-x-22-1-x-30-is
slug: the-coefficient-of-x-5-in-the-expansion-of-1-x-21-1-x-22-1-x-30-is-17035
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup></math> in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>22</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>30</mn></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>51</mn></mmultiscripts></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>9</mn></mmultiscripts></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>31</mn></mmultiscripts><mo>-</mo><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>21</mn></mmultiscripts></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>30</mn></mmultiscripts><mo>+</mo><mmultiscripts><mi>C</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>20</mn></mmultiscripts></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>31</mn></mmultiscripts><mo>-</mo><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>21</mn></mmultiscripts></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>22</mn></msup><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>30</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>1</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>9</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup><mfenced close="]" open="["><mfrac><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>10</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mfenced close="]" open="["><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>31</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup></math> in the given expression</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>x</mi></mfrac><mfenced close="]" open="["><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>31</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>6</mn></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>31</mn></msup><mo>-</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>21</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>31</mn></mmultiscripts><mo>-</mo><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>21</mn></mmultiscripts></math></p></div>


title: the-coefficient-of-x-20-in-the-expansion-of-1-x-2-40-x-2-2-x-2-1-5-is
slug: the-coefficient-of-x-20-in-the-expansion-of-1-x-2-40-x-2-2-x-2-1-5-is-51958
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of $x^{20}$ in the expansion of
<br/>$\left(1+x^{2}\right)^{40} \cdot\left(x^{2}+2+\frac{1}{x^{2}}\right)^{-5}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{30} \mathrm{C}_{10}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">${ }^{30} \mathrm{C}_{25}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{30} \mathrm{C}_{25}$</span> </div>

answer:
<div class="solution">Expression $=\left(1+x^{2}\right)^{40} \cdot\left(x+\frac{1}{x}\right)^{-10}$
<br/>$=\left(1+x^{2}\right)^{30} \cdot x^{10}$
<br/>The coefficient of $x^{20}$ in $x^{10}\left(1+x^{2}\right)^{30}$ $=$ the coefficient of $\mathrm{x}^{10}$ in $\left(1+\mathrm{x}^{2}\right)^{30}$
<br/>$={ }^{30} \mathrm{C}_{5}={ }^{30} \mathrm{C}_{30-5}={ }^{30} \mathrm{C}_{25}$</div>


title: the-coefficient-of-x-n-in-1-x-n-1-1-x-n-is
slug: the-coefficient-of-x-n-in-1-x-n-1-1-x-n-is-63434
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mi>n</mi></mrow></msup></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mi>n</mi></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>n</mi></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math></span> </li> </ul>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mi>n</mi></msup><mo>=</mo><msup><mfenced close="]" open="["><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced></mrow></mfenced><mi>n</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mi>n</mi></msup><mo>=</mo><msup><mfenced close="]" open="["><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mi>n</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mi>n</mi></msup><mo>=</mo><msup><mfenced close="]" open="["><mrow><mfenced><mrow><mn>2</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mi>n</mi></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><mi>n</mi></msup><mo>=</mo><mmultiscripts><mi>C</mi><mn>0</mn><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>·</mo><mfenced><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>·</mo><msup><mfenced><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mn>1</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><mmultiscripts><mi>C</mi><mi>n</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mo>·</mo><mfrac><mn>1</mn><msup><mi>x</mi><mi>n</mi></msup></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mrow><mo>-</mo><mi>n</mi></mrow></msup><mo>=</mo><mmultiscripts><mi>C</mi><mi>n</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts><mo>=</mo><mn>1</mn></math></p></div>


title: the-coefficient-of-x32-in-the-expansion-of-leftx4frac1x3right15-is
slug: the-coefficient-of-x32-in-the-expansion-of-leftx4frac1x3right15-is-70906
tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
question:
<div class="question">The coefficient of \(x^{32}\) in the expansion of \(\left(x^4-\frac{1}{x^3}\right)^{15}\) is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\({ }^{-15} \mathrm{C}_3\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\({ }^{15} \mathrm{C}_4\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\({ }^{-15} \mathrm{C}_5\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\({ }^{15} \mathrm{C}_2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\({ }^{15} \mathrm{C}_4\)</span> </div>

answer:
<div class="solution">We know by Binomial expansion, that \((x+a)^{\mathrm{n}}\)<br/>\(\begin{aligned}<br/>&amp;={ }^{\mathrm{n}} \mathrm{C}_0 x^{\mathrm{n}} \mathrm{a}^0+{ }^{\mathrm{n}} \mathrm{C}_1 \mathrm{x}^{\mathrm{n}-1} \cdot \mathrm{a}+{ }^{\mathrm{n}} \mathrm{C}_2 \mathrm{x}^{\mathrm{n}-2} \mathrm{a}^2 \\<br/>&amp;+{ }^{\mathrm{n}} \mathrm{C}_3 \mathrm{x}^{\mathrm{n}-3} \mathrm{a}^3+{ }^{\mathrm{n}} \mathrm{C}_4 \mathrm{x}^{\mathrm{n}-4} \cdot \mathrm{a}^4+\ldots . .+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}} \mathrm{x}^0 a^{\mathrm{n}}<br/>\end{aligned}\)<br/>Given expansion is \(\left(x^4-\frac{1}{x^3}\right)^{15}\)<br/>On comparing we get \(\mathrm{n}=15, \mathrm{x}=\mathrm{x}^4\),<br/>\(a=\left(-\frac{1}{x^3}\right)\)<br/>\(\begin{aligned}<br/>&amp; \therefore\left(x^4-\frac{1}{x^3}\right)^{15}={ }^{15} \mathrm{C}_0\left(\mathrm{x}^4\right)^{15}\left(-\frac{1}{\mathrm{x}^3}\right)^0 \\<br/>&amp; +{ }^{15} \mathrm{C}_1\left(\mathrm{x}^4\right)^{14}\left(-\frac{1}{\mathrm{x}^3}\right)+{ }^{15} \mathrm{C}_2\left(\mathrm{x}^4\right)^{13}\left(-\frac{1}{\mathrm{x}^3}\right)^2 \\<br/>&amp; +{ }^{15} \mathrm{C}_3\left(\mathrm{x}^4\right)^{12}\left(-\frac{1}{\mathrm{x}^3}\right)^3+{ }^{15} \mathrm{C}_4\left(\mathrm{x}^4\right)^{11}\left(-\frac{1}{\mathrm{x}^3}\right)^4+\ldots \\<br/>&amp; \mathrm{T}_{\mathrm{r}+1}={ }^{15} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^4\right)^{15-\mathrm{r}} \cdot\left(-\frac{1}{\mathrm{x}^3}\right)^{\mathrm{r}}=-{ }^{15} \mathrm{C}_{\mathrm{r}} \mathrm{x}^{60-7 \mathrm{r}} \\<br/>&amp; \Rightarrow \mathrm{x}{ }^{60-7 \mathrm{r}}=\mathrm{x}^{32} \Rightarrow 60-7 \mathrm{r}=32 \\<br/>&amp; \Rightarrow 7 \mathrm{r}=28 \Rightarrow \mathrm{r}=4<br/>\end{aligned}\)<br/>So, 5 th term, contains \(\mathrm{x}^{32}\)<br/>\(={ }^{15} C_4\left(x^4\right){ }^{11}\left(-\frac{1}{x^3}\right)^4={ }^{15} C_4 x^{44} x^{-12}\)<br/>\(={ }^{15} \mathrm{C}_4 \mathrm{x}^{32} .\)<br/>Thus, coefficient of \(\mathrm{x}^{32}={ }^{15} \mathrm{C}_4\).</div>


title: the-color-of-cocl-3-5-nh-3-h-2-o-is
slug: the-color-of-cocl-3-5-nh-3-h-2-o-is-91081
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">pink</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The color of $\mathrm{CoCl}_{3} .5 \mathrm{NH}_{3} \cdot \mathrm{H}_{2} \mathrm{O}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">red</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">orange</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">orange - yellow</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">pink</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\mathrm{CoCl}_{3} .5 \mathrm{NH}_{3} \mathrm{H}_{2} \mathrm{O}$ is pink in colour</div>


title: the-combination-of-gates-shown-below-yields-2
slug: the-combination-of-gates-shown-below-yields-2-71011
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The combination of gates shown below yields
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/EWo1Hg-0repPdxILuFh0Yk--LVUav_v0Q1Eqs-uv1iE.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">OR gate</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">NOT gate</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">XOR gate</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">NAND gate</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">OR gate</span> </div>

answer:
<div class="solution">The final boolean expression is,
<br/> <br/>$$
<br/> <br/>X=\overline{(\bar{A} \cdot \bar{B})}=\overline{\bar{A}}+\overline{\bar{B}}=A+B \Rightarrow \text { OR gate }
<br/> <br/>$$</div>


title: the-common-name-of-k-ptcl-3-2-c-2-h-4-is
slug: the-common-name-of-k-ptcl-3-2-c-2-h-4-is-60440
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
question:
<div class="question">The common name of $\mathrm{K}\left[\mathrm{PtCl}_{3}\left(\eta^{2} \cdot \mathrm{C}_{2} \mathrm{H}_{4}\right)\right]$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">potassium salt</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Zeise's salt</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">complex salt</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Zeise's salt</span> </div>

answer:
<div class="solution">Zeise's salt is common name of $\mathrm{K}\left[\mathrm{P} \mathrm{C}_{3}\left(\mathrm{\eta}^{2}=\mathrm{C}_{2} \mathrm{H}_{4}\right)\right]$</div>


title: the-complex-number-z-z-i-y-which-satisfies-the-equation-z-3-i-z-3-i-1-lies-on
slug: the-complex-number-z-z-i-y-which-satisfies-the-equation-z-3-i-z-3-i-1-lies-on-80309
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The complex number $z=z+i y$ which satisfies
<br/> <br/>the equation $\left|\frac{z-3 i}{z+3 i}\right|=1,$ lies on</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">the $X$ -axis</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">the straight line $y=3$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a circle passing through origin</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the $X$ -axis</span> </div>

answer:
<div class="solution">$\left|\frac{z-3 i}{z+3 i}\right|=1 \Rightarrow|z-3 i|=|z+3 i|$
<br/> <br/>[if $\left|z-z_{1}\right|=\left|z+z_{2}\right|,$ then it is a perpendicular bisector of $z_{1}$ and $z_{2}$ ] Hence, perpendicular bisector of (0,3) and (0,-3) is $X$ -axis.</div>


title: the-compound-which-gives-the-most-stable-carbonium-ion-on-dehydration-is
slug: the-compound-which-gives-the-most-stable-carbonium-ion-on-dehydration-is-22360
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The compound which gives the most stable carbonium ion on dehydration is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CH}\left(\mathrm{CH}_{3}\right) \mathrm{CH}_{2} \mathrm{OH}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CH}_{2}=\mathrm{CHCH}_{2} \mathrm{CH}_{2} \mathrm{OH}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CHOHCH}_{2} \mathrm{CH}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\mathrm{CH}_{3}\right)_{3} \mathrm{COH}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/UbwmVFeMm7GyWps8b-N0zzVW1cPxIOChaYKg7qLCan4.original.fullsize.png"/></div>


title: the-compounds-ptcl-2-nh-3-4-br-2-and-ptbr-2-nh-3-4-cl-2-constitutes-a-pair-of
slug: the-compounds-ptcl-2-nh-3-4-br-2-and-ptbr-2-nh-3-4-cl-2-constitutes-a-pair-of-25094
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The compounds $\left[\mathrm{PtCl}_2\left(\mathrm{NH}_3\right)_4\right] \mathrm{Br}_2$ and $\left[\mathrm{PtBr}_2\left(\mathrm{NH}_3\right)_4\right] \mathrm{Cl}_2$ constitutes a pair of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>coordination isomers</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>linkage isomers</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>ionization isomers</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>optical isomers</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>ionization isomers</span> </div>

answer:
<div class="solution">$\left[\mathrm{PtCl}_2\left(\mathrm{NH}_3\right)_4\right] \mathrm{Br}_2$ and $\left[\mathrm{PtBr}_2\left(\mathrm{NH}_3\right)_4\right] \mathrm{Cl}_2$ are ionisation isomers.</div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-compressibility-of-water-is-4-1-0-5-per-unit-atmospheric-pressure-the-decrease-in-volume-of-100-cm-3-of-water-under-a-pressure-of-100-atmosphere-w

slug:
the-compressibility-of-water-is-4-1-0-5-per-unit-atmospheric-pressure-the-decrease-in-volume-of-100-cm-3-of-water-under-a-pressure-of-100-atmosphere-w-28975

question:
<div class="question">The compressibility of water is $4 \times 10^{-5}$ per unit atmospheric pressure. The decrease in volume of $100 \mathrm{~cm}^{3}$ of water under a pressure of 100 atmosphere will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.4 \mathrm{~cm}^{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4 \times 10^{-5} \mathrm{~cm}^{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.025 \mathrm{~cm}^{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.004 \mathrm{~cm}^{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.4 \mathrm{~cm}^{3}$</span> </div>

answer:
<div class="solution">Compressibility $=\frac{1}{\mathrm{~B}}=\frac{\Delta \mathrm{V} / \mathrm{V}}{\mathrm{P}}$.
<br/> <br/>Here, $\mathrm{P}=100 \mathrm{~atm}$
<br/> <br/>Compressibility $=4 \times 10^{-5}$ and $\mathrm{V}=100 \mathrm{~cm}^{3}$
<br/> <br/>Hence, $\Delta \mathrm{V}=0.4 \mathrm{~cm}^{3}$</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
title:
the-conductivity-of-a-saturated-solution-of-baso-4-is-306-1-0-6-ohm-1-cm-1-and-its-equivalent-conductance-is-153-ohm-1-cm-2-equiv-1-the-k-sp-for-baso-

slug:
the-conductivity-of-a-saturated-solution-of-baso-4-is-306-1-0-6-ohm-1-cm-1-and-its-equivalent-conductance-is-153-ohm-1-cm-2-equiv-1-the-k-sp-for-baso-79949

question:
<div class="question">The conductivity of a saturated solution of $\mathrm{BaSO}_{4}$ is $3.06 \times 10^{-6} \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}$ and its equivalent conductance is $1.53 \mathrm{ohm}^{-1} \mathrm{~cm}^{2}$ equiv $^{-1}$. The $\mathrm{K}_{\mathrm{sp}}$ for $\mathrm{BaSO}_{4}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 \times 10^{-12}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.5 \times 10^{-9}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.5 \times 10^{-13}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 \times 10^{-6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \times 10^{-6}$</span> </div>

answer:
<div class="solution">Solublity $=\frac{\text { conductivity } \times 1000}{\Lambda_{\text {eq }}}$<br/>$$<br/>\begin{array}{l}<br/>=\frac{3.06 \times 10^{-6} \times 1000}{1.53}=2 \times 10^{-3} \\<br/>K_{\text {sp }}=\mathrm{S}^{2}=4 \times 10^{-6}<br/>\end{array}<br/>$$</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">12</span> </div>
source_db: MarksBatch2_P2.db
title:
the-conductivity-of-a-weak-acid-ha-of-concentration-0001-mol-l-1-is-20-1-0-5-s-cm-1-if-m-ha-190-s-cm-2-mol-1-the-ionization-constant-k-a-of-ha-is-equa

slug:
the-conductivity-of-a-weak-acid-ha-of-concentration-0001-mol-l-1-is-20-1-0-5-s-cm-1-if-m-ha-190-s-cm-2-mol-1-the-ionization-constant-k-a-of-ha-is-equa-17511

question:
<div class="question">The conductivity of a weak acid HA of concentration $0.001 \mathrm{~mol} \mathrm{~L}^{-1}$ is $2.0 \times 10^{-5} \mathrm{~S} \mathrm{~cm}^{-1}$.<br/>If $\Lambda_{\mathrm{m}}^{\circ}(\mathrm{HA})=190 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$, the ionization constant $\left(\mathrm{K}_{\mathrm{a}}\right)$ of $\mathrm{HA}$ is equal to _______ x $10^{-6}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">24</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">12</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">45</span> </li> </ul>

answer:
<div class="solution">$\Lambda_{\mathrm{m}}=1000 \times \frac{\mathrm{K}}{\mathrm{M}}$<br/>$=1000 \times \frac{2 \times 10^{-5}}{0.001}=20 \mathrm{~S} \mathrm{~cm}^2 \mathrm{~mol}^{-1}$<br/>$\Rightarrow \alpha=\frac{\Lambda_{\mathrm{m}}}{\Lambda_{\mathrm{m}}^{\circ}}=\frac{20}{190}=\left(\frac{2}{19}\right)$<br/>$\underset{0.001(1-\alpha)}{\mathrm{HA}} \rightleftharpoons \underset{0.001 \alpha}{\mathrm{H}^{+}}+\underset{0.001 \alpha}{\mathrm{A}^{-}}$<br/>$K_{\mathrm{a}}=0.001\left(\frac{\alpha^2}{1-\alpha}\right)=\frac{0.001 \times\left(\frac{2}{19}\right)^2}{1-\left(\frac{2}{19}\right)}$<br/>$=12.3 \times 10^{-6}$</div>


title: the-cooking-time-in-a-pressure-cooker-is-less-because
slug: the-cooking-time-in-a-pressure-cooker-is-less-because-36126
question: <div class="question">The cooking time in a pressure cooker is less because:</div>
tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">More heat is used</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">High pressure cooks the food</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The boiling point of water increases in the cooker</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Heat is uniformly distributed For the reaction $: \mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The boiling point of water increases in the cooker</span> </div>

answer:
<div class="solution">In pressure cooker, pressure is high thus, the boiling point of water increases, resulting cooking time is less than other open pots.</div>


title: the-coordinates-of-the-foot-of-the-perpendicular-from-the-point-2-3-on-the-line-x-y-11-0-are
slug: the-coordinates-of-the-foot-of-the-perpendicular-from-the-point-2-3-on-the-line-x-y-11-0-are-72102
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The coordinates of the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$ are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>(-6,5)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>(5,6)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>(-5,6)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>(6,5)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>(5,6)</span> </div>

answer:
<div class="solution">Let $(h, k)$ be the coordinates of the foot of the perpendicular from the point $(2,3)$ on the $\mathrm{i}$ $x+y-11=0$. Then, the slope of the perpendicular line is $\frac{k-3}{h-2}$. Again the slope of the given line $x+y-11=0$ is -1<br/>Using the condition of perpendicularity of lines, we have<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/C-WbwInk0aGjGqJQDt1Issw2u3kiXNeGDvlnlsECidE.original.fullsize.png"/><br/><br/>Since $(h, k)$ lies on the given line, we have,<img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/pT2Obwjxx5U43qmQ_4CYo1Ue5AOKW0NWjysa93VVUVs.original.fullsize.png"/><br/><br/>Solving (i) and (ii), we get $h=5$ and $k=6$. Thus $(5,6)$ are the required coordinates of the foot of the perpendicular.</div>


tags: ['Mathematics', 'Linear Programming', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-coordinates-of-the-point-at-which-minimum-value-of-z-7-x-8-y-subject-to-constraints-x-y-20-0-y-5-x-0-y-0-is-attained-is

slug:
the-coordinates-of-the-point-at-which-minimum-value-of-z-7-x-8-y-subject-to-constraints-x-y-20-0-y-5-x-0-y-0-is-attained-is-58898

question:
<div class="question">The coordinates of the point at which minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>7</mn><mi>x</mi><mo>-</mo><mn>8</mn><mi>y</mi></math> subject to constraints <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>-</mo><mn>20</mn><mo>≤</mo><mn>0</mn><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>≥</mo><mn>5</mn><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mn>0</mn><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>≥</mo><mn>0</mn></math> is attained, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>20</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>15</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>20</mn></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>20</mn></mrow></mfenced></math></span> </div>

answer:
<div class="solution">Feasible region is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mi>C</mi><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>7</mn><mi>x</mi><mo>-</mo><mn>8</mn><mi>y</mi></math><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/29ad1c2a-a79d-4dee-9be2-09892ed13180-image.png" style="width: 370px; height: 264px;"/></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mfenced><mrow><mn>0</mn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>Z</mi><mo>=</mo><mn>7</mn><mo>×</mo><mn>0</mn><mo>-</mo><mn>8</mn><mo>×</mo><mn>5</mn><mo>=</mo><mo>-</mo><mn>40</mn></math></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mfenced><mrow><mn>15</mn><mo>,</mo><mn>5</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>Z</mi><mo>=</mo><mn>7</mn><mfenced><mn>15</mn></mfenced><mo>-</mo><mn>8</mn><mfenced><mn>5</mn></mfenced><mo>=</mo><mn>65</mn></math></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mfenced><mrow><mn>0</mn><mo>,</mo><mn>20</mn></mrow></mfenced><mo>,</mo><mo> </mo><mi>Z</mi><mo>=</mo><mn>7</mn><mfenced><mn>0</mn></mfenced><mo>-</mo><mn>8</mn><mfenced><mn>20</mn></mfenced><mo>=</mo><mo>-</mo><mn>160</mn></math></p><p>Hence, the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math> is attained at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>20</mn></mrow></mfenced></math></p><p>So, option <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>d</mi></mfenced></math> is correct.</p></div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
title:
the-coordinates-of-the-point-where-the-line-through-the-points-a-3-4-1-and-b-5-1-6-crosses-the-xyplane-are

slug:
the-coordinates-of-the-point-where-the-line-through-the-points-a-3-4-1-and-b-5-1-6-crosses-the-xyplane-are-58063

question:
<div class="question">The coordinates of the point where the line through the points $\mathrm{A}(3,4,1)$ and $\mathrm{B}(5,1,6)$ crosses the XY-plane are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\frac{13}{5}, \frac{23}{5}, 0\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(-\frac{13}{5}, \frac{23}{5}, 0\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{13}{5},-\frac{23}{5}, 0\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(-\frac{13}{5}, \frac{-23}{5}, 0\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{13}{5}, \frac{23}{5}, 0\right)$</span> </div>

answer:
<div class="solution">Equation of the line through the given points is $\frac{x-3}{5-3}=\frac{y-4}{1-4}=\frac{z-1}{6-1}$
<br/> <br/>$\Rightarrow \frac{x-3}{2}=\frac{y-4}{-3}=\frac{z-1}{5}$
<br/> <br/>Any point on this line can be taken as $(3+$ $2 \lambda, 4-3 \lambda, 1+5 \lambda)$
<br/> <br/>If this point lies on XY-plane then the z-coordinate is zero
<br/> <br/>$$
<br/> <br/>\Rightarrow 1+5 \lambda=0 \Rightarrow \lambda=-\frac{1}{5}
<br/> <br/>$$
<br/> <br/>Thus the required coordinates of the point are
<br/> <br/>$$
<br/> <br/>\left(3-\frac{2}{5}, 4-3\left(-\frac{1}{5}\right), 0\right) \equiv\left(\frac{13}{5}, \frac{23}{5}, 0\right)
<br/> <br/>$$</div>


title: the-correct-bond-order-in-the-following-species-is-1
slug: the-correct-bond-order-in-the-following-species-is-1-78821
question: <div class="question">The correct bond order in the following species is:</div>
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{O}_{2}^{2+}&lt;\mathrm{O}_{2}^{-}&lt;\mathrm{O}_{2}^{+}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{O}_{2}^{+}&lt;\mathrm{O}_{2}^{-}&lt;\mathrm{O}_{2}^{2+}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{O}_{2}^{-}&lt;\mathrm{O}_{2}^{+}&lt;\mathrm{O}_{2}^{2+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{O}_{2}^{2+}&lt;\mathrm{O}_{2}^{+}&lt;\mathrm{O}_{2}^{-}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{O}_{2}^{-}&lt;\mathrm{O}_{2}^{+}&lt;\mathrm{O}_{2}^{2+}$</span> </div>

answer:
<div class="solution">$\mathrm{O}_{2}^{+}$ ion - Total number of electrons<br/>$(16-1)=15$<br/>E.C.:<br/>$$<br/>\begin{array}{r}<br/>\sigma 1 s^{2} \sigma^{*} 1 s^{2} \sigma 2 s^{2} \sigma^{*} 2 s^{2} \sigma 2 p_{z}^{2} \\<br/>\pi 2 p_{x}^{2}=\pi 2 p_{y}^{2} \pi^{*} 2 p_{x}^{1}=\pi^{*} 2 p_{y} \\<br/>\text { Bond order }=\frac{\mathrm{N}_{\mathrm{b}}-\mathrm{N}_{\mathrm{a}}}{2}=\frac{10-5}{2}=\frac{5}{2}=2 \frac{1}{2}<br/>\end{array}<br/>$$<br/>$\mathrm{O}_{2}^{-}$ (Super oxide ion): Total number of<br/>electrons $(16+1)=17$<br/>E.C.:<br/>$$<br/>\begin{array}{l}<br/>\text { \sigma1s }^{2} \sigma^{*} 1 s^{2} \sigma 2 s^{2} \sigma^{*} 2 s^{2} \sigma 2 p_{z}^{2} \\<br/>\pi 2 p_{x}^{2}=\pi 2 p_{y}^{2} \pi^{*} 2 p_{x}^{2}=\pi^{*} 2 p_{y}^{1} \\<br/>\text { Bond order }=\frac{\left(\mathrm{N}_{\mathrm{b}}-\mathrm{N}_{\mathrm{a}}\right)}{2}=\frac{10-7}{2}=\frac{3}{2}=1 \frac{1}{2}<br/>\end{array}<br/>$$<br/>$\mathrm{O}_{2}^{2+}$ ion: Total number of electrons $=(16-2)=14$<br/>E.C.<br/>$$<br/>\begin{array}{l}<br/>\sigma 1 s^{2} \sigma^{*} 1 s^{2} \sigma 2 s^{2} \sigma^{*} 2 s^{2} \pi 2 p_{x}^{2}=\pi 2 p_{y}^{2} \sigma 2 p_{z}^{2} \\<br/>\text { Bond order }=\frac{\left(N_{b}-N_{a}\right)}{2}=\frac{10-4}{2}=\frac{6}{2}=3<br/>\end{array}<br/>$$<br/>So bond order: $\mathrm{O}_{2}^{-}&lt;\mathrm{O}_{2}^{+}&lt;\mathrm{O}_{2}^{2+}$</div>


title: the-correct-nucleophilicity-order-is
slug: the-correct-nucleophilicity-order-is-91339
question: <div class="question">The correct nucleophilicity order is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CH}_{3}^{-}&lt;\mathrm{NH}_{2}^{-}&lt;\mathrm{HO}^{-}&lt;\mathrm{F}^{-}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CH}_{3}^{-} \simeq \mathrm{NH}_{2}^{-}&gt;\mathrm{HO}^{-} \simeq \mathrm{F}^{-}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{CH}_{3}^{-}&gt;\mathrm{NH}_{2}^{-}&gt;\mathrm{HO}^{-}&gt;\mathrm{F}^{-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{NH}_{2}^{-}&gt;\mathrm{F}^{-}&gt;\mathrm{HO}^{-}&gt;\mathrm{CH}_{3}^{-}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CH}_{3}^{-}&gt;\mathrm{NH}_{2}^{-}&gt;\mathrm{HO}^{-}&gt;\mathrm{F}^{-}$</span> </div>

answer:
<div class="solution">Nucleophilicity increases with the decrease in electronegativity of the central atom. Since electronegativity follows the order:<br/>$\mathrm{F}&gt;\mathrm{O}&gt;\mathrm{N}&gt;\mathrm{C} ;$ nucleophilicity of the<br/>concerned group will follow the reverse order i.e.,<br/>$\mathrm{CH}_{3}^{-}&gt;\mathrm{NH}_{2}^{-}&gt;\mathrm{OH}^{-}&gt;\mathrm{F}^{-}$</div>


title: the-correct-order-of-acidic-strength-1
slug: the-correct-order-of-acidic-strength-1-59628
question: <div class="question">The correct order of acidic strength :</div>
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{Cl}_{2} \mathrm{O}_{7}&gt;\mathrm{SO}_{2}&gt;\mathrm{P}_{4} \mathrm{O}_{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{K}_{2} \mathrm{O}&gt;\mathrm{CaO}&gt;\mathrm{MgO}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CO}_{2}&gt;\mathrm{N}_{2} \mathrm{O}_{5}&gt;\mathrm{SO}_{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Na}_{2} \mathrm{O}&gt;\mathrm{MgO}&gt;\mathrm{Al}_{2} \mathrm{O}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Cl}_{2} \mathrm{O}_{7}&gt;\mathrm{SO}_{2}&gt;\mathrm{P}_{4} \mathrm{O}_{10}$</span> </div>

answer:
<div class="solution">Acidic character of oxide $\propto$ Non-metallic nature of element. Non-metallic character increases along the period. Hence order of acidic character is
<br/> <br/>$\mathrm{Cl}_{2} \mathrm{O}_{7}&gt;\mathrm{SO}_{2}&gt;\mathrm{P}_{4} \mathrm{O}_{10}$</div>


title: the-correct-order-of-increasing-c-o-bond-length-co-co-3-2-co-2-is-1
slug: the-correct-order-of-increasing-c-o-bond-length-co-co-3-2-co-2-is-1-15173
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">The correct order of increasing $\mathrm{C}-\mathrm{O}$ bond length $\mathrm{CO}, \mathrm{CO}_{3}^{2-}, \mathrm{CO}_{2}$ is:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{CO}&lt;\mathrm{CO}_{2}&lt;\mathrm{CO}_{3}^{2-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CO}_{2}&lt;\mathrm{CO}_{3}^{2-}&lt;\mathrm{CO}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CO}&lt;\mathrm{CO}_{3}^{2-}&lt;\mathrm{CO}_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CO}_{3}^{2-}&lt;\mathrm{CO}_{3}&lt;\mathrm{CO}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CO}&lt;\mathrm{CO}_{2}&lt;\mathrm{CO}_{3}^{2-}$</span> </div>

answer:
<div class="solution">$\mathrm{C} \equiv \mathrm{Ohas}$ lowest bond length due to highest bond order. $\mathrm{O}=\mathrm{C}=\mathrm{O}$ has second lowest bond length due to double bond.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/59NZmD41nCRFHd9WGd8a4BKP7zFCShzEp5Ag_LE5u-I.original.fullsize.png"/><br/> <br/> <br/>has highest has bond length due to lowest bond order which is due to
<br/> <br/>resonance.</div>


title: the-correct-order-of-magnetic-moments-spin-only-values-in-bm-is
slug: the-correct-order-of-magnetic-moments-spin-only-values-in-bm-is-74287
question: <div class="question">The correct order of magnetic moments (spin only values in B.M.) is:</div>
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}&gt;\left[\mathrm{MnCl}_{4}\right]^{2-}&gt;\left[\mathrm{CoCl}_{4}\right]^{2-}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{MnCl}_{4}\right]^{2-}&gt;\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}&gt;\left[\mathrm{CoCl}_{4}\right]^{2-}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{MnCl}_{4}\right]^{2-}&gt;\left[\mathrm{CoCl}_{4}\right]^{2-}&gt;\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}&gt;\left[\mathrm{CoCl}_{4}\right]^{2-}&gt;\left[\mathrm{MnCl}_{4}\right]^{2-}$
<br/> <br/>(Atomic nos. : $\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{MnCl}_{4}\right]^{2-}&gt;\left[\mathrm{CoCl}_{4}\right]^{2-}&gt;\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}$</span> </div>

answer:
<div class="solution">Number of unpaired electrons in central atom
<br/> <br/>$$
<br/> <br/>\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-},\left[\mathrm{CoCl}_{4}\right]^{2-},\left[\mathrm{MnCl}_{4}\right]^{2-}
<br/> <br/>$$
<br/> <br/>zer three five The greater the number of unpaired electrons, the higher the value of magnetic
<br/> <br/>moment</div>


title: the-correct-order-of-mobility-of-alkali-metal-ions-in-aqueous-solution-is
slug: the-correct-order-of-mobility-of-alkali-metal-ions-in-aqueous-solution-is-67223
tags: ['Chemistry', 's Block Elements', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">The correct order of mobility of alkali metal ions in aqueous solution is-</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Li</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Na</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi mathvariant="normal">K</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Rb</mi><mo>+</mo></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Na</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi mathvariant="normal">K</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Rb</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Li</mi><mo>+</mo></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">K</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Rb</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Na</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Li</mi><mo>+</mo></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Rb</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi mathvariant="normal">K</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Na</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Li</mi><mo>+</mo></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Rb</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi mathvariant="normal">K</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Na</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Li</mi><mo>+</mo></msup></math></span> </div>

answer:
<div class="solution"><p>Mobility of alkali ions in aqueous solution <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∝</mo><mfrac><mn>1</mn><mtext> size of hydrated ion </mtext></mfrac></math></p><p>The mobility of alkali ions is inversely proportional to the size of ion in hydrated state, because bigger ion can't move faster. Also, small ion get hydrolysed to large exent due to which its size get increased. In other words, smaller ion will form hydrated ion of larger size.</p><p>Hence, correct order of mobility is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Rb</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi mathvariant="normal">K</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Na</mi><mo>+</mo></msup><mo>&gt;</mo><msup><mi>Li</mi><mo>+</mo></msup></math></p></div>


title: the-correct-order-of-radii-is
slug: the-correct-order-of-radii-is-21081
question: <div class="question">The correct order of radii is</div>
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{N}&lt;\mathrm{Be}&lt;\mathrm{B}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{F}^{-}&lt;\mathrm{O}^{2-}&lt;\mathrm{N}^{3-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{Na}&lt;\mathrm{L}_{1}&lt;\mathrm{K}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Fe}^{3+}&lt;\mathrm{Fe}^{2+}&lt;\mathrm{Fe}^{4+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{F}^{-}&lt;\mathrm{O}^{2-}&lt;\mathrm{N}^{3-}$</span> </div>

answer:
<div class="solution">Effective nuclear charge (i,e. Z/e ratio) decreases from $\mathrm{F}^{-}$ to $\mathrm{N}^{3-}$ hence the radii follows the order: $\mathrm{F}^{-}&lt;\mathrm{O}^{2-}&lt;\mathrm{N}^{3-} . \mathrm{Z} / \mathrm{e}$ for<br/>$\mathrm{F}^{-}=9 / 10=0.9,$ for $\mathrm{O}^{2-}=8 / 10=.8,$ for $\mathrm{N}^{3-}=0.7$</div>


title: the-correct-order-of-reactivity-of-halogens-with-alkalies-is
slug: the-correct-order-of-reactivity-of-halogens-with-alkalies-is-26528
question: <div class="question">The correct order of reactivity of halogens with alkalies is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{F}&gt;\mathrm{Cl}&gt;\mathrm{Br}&gt;\mathrm{I}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{F}&lt;\mathrm{Cl}&gt;\mathrm{Br}&lt;\mathrm{I}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{F}&lt;\mathrm{Cl}&lt;\mathrm{Br}&lt;\mathrm{I}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad \mathrm{F}&lt;\mathrm{Cl}&lt;\mathrm{Br}&gt;\mathrm{I}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \mathrm{F}&gt;\mathrm{Cl}&gt;\mathrm{Br}&gt;\mathrm{I}$</span> </div>

answer:
<div class="solution">Reactivity follows the order $\mathrm{F}&gt;\mathrm{Cl}&gt;\mathrm{Br}&gt;\mathrm{I}$</div>


title: the-correct-sequence-of-reagents-for-the-following-conversion-will-be-1
slug: the-correct-sequence-of-reagents-for-the-following-conversion-will-be-1-68877
tags: ['Chemistry', 'Hydrocarbons', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question"><p>The correct sequence of reagents for the following conversion will be</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/545e2c13-ad63-4a89-b274-13946448bec8-image.png" style="width: 474px; height: 120px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub><mo>/</mo><mi>Red</mi><mo> </mo><mi mathvariant="normal">P</mi><mo>,</mo><mo> </mo><msub><mi>AlCl</mi><mn>3</mn></msub><mo>,</mo><mo> </mo><mo> </mo><mi>MeCOOH</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><msub><mi>SO</mi><mn>4</mn></msub><mo>+</mo><msub><mi>HgSO</mi><mn>4</mn></msub><mo>,</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>/</mo><mi>Heat</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">O</mi><mn>3</mn></msub><mo>/</mo><mi>Zn</mi><mo>-</mo><mi>AcOH</mi><mo>,</mo><mo> </mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><msub><mi>SO</mi><mn>4</mn></msub><mo>+</mo><msub><mi>HgSO</mi><mn>4</mn></msub><mo>/</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>/</mo><mi>Heat</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mi>COOH</mi><mo>,</mo><mo> </mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mo>+</mo><mi>OH</mi><mo>/</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><msub><mi>SO</mi><mn>4</mn></msub><mo>+</mo><msub><mi>HgSO</mi><mn>4</mn></msub><mo>,</mo><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>/</mo><mi>Heat</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo>-</mo><mi mathvariant="normal">C</mi><mo>≡</mo><mi>CH</mi></math> is converted to ketone by catalytic hydration with reagents.</p><p>Hydration of alkynes is carried out in the presence of a warm 40% sulfuric acid in the presence of 1% mercuric sulfate.</p><p>In this reaction, first a molecule of water is added to form enol which then tautomerizes to keto form. </p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e34665d5-bd51-4277-93ad-ac699119b381-image.png" style="width: 524px; height: 100px;"/></p></div>


title: the-correct-stability-order-of-following-species-is-1
slug: the-correct-stability-order-of-following-species-is-1-70283
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The correct stability order of following species is -
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Xlowm4C2XksQH3G6bi5ZpKxnx4FnZGhKQT_c7R6S6YA.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3Y6XCSSB-KZxD-6ieOsWnCwNaKWOhWzYaRPOXaeHtUw.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/FkObPo5-iP0hWPi4_F79iaMjUPSb9j2E4AU80S1DtsE.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/xwxkwsN3G8RWy52SJWHForKEPBy1PsLDTZBycKdmsIM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x&gt;y&gt;w&gt;z$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y&gt;x&gt;w&gt;z$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x&gt;w&gt;z&gt;y$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$z&gt;x&gt;y&gt;w$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x&gt;w&gt;z&gt;y$</span> </div>

answer:
<div class="solution">$x$ has a conjugated diene system, $\mathrm{w}$ an isolated diene system, $\mathrm{z}$ a cumulated diene system, and y an antiaromatic system.</div>


title: the-correct-statement-about-silicone-is-1
slug: the-correct-statement-about-silicone-is-1-30800
question: <div class="question">The correct statement about silicone is</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'JEE Main']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">They are ketones with silyl group <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>SiH</mi><mn>3</mn></msub></mfenced></math> similar to alkyl, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><msub><mi>SiH</mi><mn>3</mn></msub></mfenced><mn>2</mn></msub><mi>CO</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">They are synthetic polymer containing repeated <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><msub><mi>SiO</mi><mn>2</mn></msub><mo> </mo><mi>units</mi></math>.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">They are formed by hydrolysis of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><msub><mi>SiCl</mi><mn>2</mn></msub></math>.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">They are formed by hydrolysis of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><msub><mi>SiCl</mi><mn>2</mn></msub></math>.</span> </div>

answer:
<div class="solution"><blockquote><p>Silicones are synthetic organo-silicon polymers containing repeated <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><mi>SiO</mi></math> units. Since the empirical formula is same as that of a ketone <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><mi>CO</mi></mrow></mfenced></math>, the name silicon has been given to these materials. They can be formed by hydrolysis of dichlorosilanes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><msub><mi>SiCl</mi><mn>2</mn></msub></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><msub><mi>SiCl</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>⟶</mo><msub><mi mathvariant="normal">R</mi><mn>2</mn></msub><mi>Si</mi><msub><mfenced><mi>OH</mi></mfenced><mn>2</mn></msub><mo>+</mo><mn>2</mn><mi>HCl</mi></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c7856c4e-79e0-4292-850d-d703c5b70342-image.png" style="width: 505px; height: 239px;"/></p></blockquote></div>


title: the-correct-statement-about-the-following-disaccharide-is-1
slug: the-correct-statement-about-the-following-disaccharide-is-1-43699
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question"><p>The correct statement about the following disaccharide is-</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e4a6db5c-58bd-4cc1-8998-c140ff07ca07-qwqwqa.png" style="width: 629px; height: 312px;"/></p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Ring <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">I</mi></mfenced></math> is pyranose with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math>-glycosidic link</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Ring <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">I</mi></mfenced></math> is furanose with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math>-glycosidic link</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Ring <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>II</mi></mfenced></math> is furanose with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math>-glycosidic link</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Ring <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>II</mi></mfenced></math> is pyranose with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math>-glycosidic link</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Ring <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">I</mi></mfenced></math> is pyranose with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math>-glycosidic link</span> </div>

answer:
<div class="solution"><p>In the given structure, ring one <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi mathvariant="normal">I</mi></mfenced></math> has one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">O</mi></math>-atom and is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> membered ring, thus is called pyranose ring, whereas ring <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>II</mi></mfenced></math> has one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">O</mi></math>-atom and is a five membered ring, thus is called furanose.</p><p>The disaccharide is linked by a glycosidic bond between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>-</mo><mn>1</mn></math> of glucose (in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math> -position, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">α</mi></math> linkage) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi><mo>-</mo><mn>2</mn></math> of fructose (in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">β</mi></math>-position, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">β</mi></math>-linkage).</p></div>


title: the-curve-given-by-x-y-e-x-y-has-a-tangent-parallel-to-the-yaxis-at-the-point
slug: the-curve-given-by-x-y-e-x-y-has-a-tangent-parallel-to-the-yaxis-at-the-point-75058
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The curve given by $x+y=e^{x y}$ has a tangent parallel to the Y-axis at the point</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(0,1)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(1,0)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(1,1)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(1,0)$</span> </div>

answer:
<div class="solution">$\because x+y=e^{x y}$<br/>Differentiating w.r.t. $x$, we get<br/>$1+\frac{d y}{d x}=e^{x y}\left[y+x \frac{d y}{d x}\right]$<br/>$\Rightarrow \quad \frac{d y}{d x}\left(1-x e^{x y}\right)=y e^{x y}-1$<br/>$\Rightarrow \frac{d y}{d x}=\frac{y e^{x y}-1}{1-x e^{x y}} \because \frac{d y}{d x}=\infty$,<br/>as tangent is parallel to $\mathrm{Y}$-axis<br/>$\Rightarrow 1-x e^{x y}=0$<br/>$\therefore \mathrm{xe}^{\mathrm{xy}}=1$<br/>This holds, when $x=1$ and $y=0$</div>


